First of all, welcome to competitive Pokemon! It's a very complex and subjective game with many intricacies and lots of room for error, but I think it's the best part of Pokemon for those exact reasons.
Before beginning, I should probably verify that this answer is mostly directed toward competitive Pokemon using Smogon rules, which is where the majority of the competitive community lies and where the most balanced and closely regulated sets of rules come from. Make sure you're familiar with Smogon's playable tiers and their battle rules; even if you choose not to associate with their competitive game and rules, it pays to gain an understanding of how they function since they're so broadly and commonly referenced. This doesn't have a huge effect on the actual types of teams and what their core concepts are, but it certainly affects which ones are the most viable and which Pokemon may be used, as wifi battles and Smogon OU do have some notable variations in rules.
If you haven't yet been introduced to Pokemon Showdown, you should know that it's easily the most convenient and accessible platform for competitive Pokemon, and runs according to Smogon's rules. While in-game competitive Pokemon certainly has merit and is being made increasingly convenient by the developers, Showdown is currently the place to be for newcomers and experienced players alike.
But to address the actual question: in competitive Pokemon, there are many different team archetypes that you can build around, each with many different variations in design and actual playstyle. There are three main ones you should be familiar with with though, which are as follows. Remember that there are no rules in teambuilding; though these team "types" generally formulaic and don't deviate from their normal structure, sometimes due to your choice of Pokemon or a weakness in your team it might be wise to incorporate something that is otherwise considered unusual.
Balanced
This playstyle is largely self explanatory in that it incorporates a mix and literal "balance" of different offensive and defensive Pokemon. Balance depends largely on the player's ability to anticipate the opponent's move and make switches into Pokemon that respond to the opponent's likely move. This play can then be used to gain an advantage, either by threatening the opponent with the Pokemon in play or making another preemptive switch that puts you in a good position. Pokemon that are used to gain this advantage can be called "pivots", and the pressure they apply to the opponent that forces them to make a defensive switch or a prediction is called "momentum", which is a term lightly applied to whenever you have rough control of the battle as the Pokemon you have in play threatens the opponent's in some way and forces them to react. Maintaining this momentum is predicting how they will react and making a play in accordance to this, which will eventually lead to a KO.
The structure of a balance team is based on these concepts. Generally, it will consist of a couple of strong offensive Pokemon that pose an immediate offensive threat to the opponent without setting up, often called "wallbreakers" or just "breakers". Typically, it is these Pokemon that put in the work and take out the opponent's team, and are the Pokemon you will play to try and win with, called a "win condition" based on your opponent's team and which Pokemon is best suited for taking it out. Then, there will be the "pivots" mentioned above, which are there to take the hits when your breakers are under threat and force the opponent to make a defensive play, allowing you to make a proactive play like an attack or make a switch back into a breaker, which will hopefully force your opponent to take a hit and maybe lose a Pokemon. Pivots aren't exclusively defensive Pokemon designed to take a hit; sometimes, they just have a defensive typing that works really well with a breaker that means they can cover each other's weaknesses, or an ability like Intimidate that allows them to switch in safely. Landorus-Therian is a good example; it resists common attacking types like Fighting and Ground, packs Intimidate, has a strong offensive presence to pressure the opponent and has moves like U-Turn that maintain momentum, making it a highly favourable pivot despite its defensive stats being fairly standard.
As balance typically involves a heavy amount of switching around due to its dependence on pivots, Pokemon that can remove popular entry hazards like Spikes and Stealth Rock are great on balance teams (Pokemon that fill this role can be called "utility" Pokemon, though hazard removers aren't exclusively defensively oriented as this would suggest). Conversely, you might also want a Pokemon that can set up said entry hazards, or a set-up sweeper Pokemon that can clean up the game when given the opportunity.
Offensive
Offence is the broadest category of teams in competitive Pokemon. Though offence teams are all based on one core concept -- gaining an advantage, or "momentum", through offensive pressure -- they come in many different playstyles that cause you to play the team differently. Offence typically requires that you make proactive, offensive plays to clean the opponent's team, in contrast to other playstyles that revolve around pivoting to gain an advantage (balance teams) and blocking your opponent's team by walling it, crippling it and applying passive damage to win (stall-based teams).
Offence is typically built with a bunch of setup sweepers and breakers -- often frail and poor defensively -- which work together using offensive synergy to defeat the opponent's Pokemon, eventually opening the door for a win condition Pokemon to secure a final sweep. In offence, the goal is (usually) not to be able to switch into different moves, but instead make it so the opponent can't switch into or somehow resist yours. This is where the concept of "offensive synergy" comes into play; if you have a set of two or three offensive Pokemon that together can break though significant potions of the metagame and set up a final sweep of an opposing team, then their defensive synergy (ability to take attacks for each other) is far less relevant, especially considering these Pokemon are often very frail and shouldn't switch into attacks anyways. Offense instead focuses on ensuring that the win condition Pokemon's direct checks and counters are either eliminated, or weakened enough to the point where they can be safely knocked out. For example, if I have a Talonflame that can sweep my opponent's team except for the Latios they have at full health that will cleanly knock Talonflame out with Draco Meteor, I should try to weaken Latios to around 70% HP so that Brave Bird will reliably knock it out, or even knock it out completely. (Use a damage calculator to figure out when Pokemon are in range of your attacks; the calculator linked here has pre-loaded movesets that Pokemon very commonly use, as well as blank Pokemon for experimentation.)
Some offensive teams, known as hyper offence teams, take things to the next level and basically incorporate sacrificing Pokemon into the their design. The typically run multiple set-up sweeping Pokemon, some breakers and entry hazard removers and setters where necessary, and sometimes a Pokemon to break stall teams. Hyper offence teams are designed and played in a linear fashion; they take one potential set-up win condition, then add another set-up win condition based on what counters the first one. In this way, hyper offence can take advantage of the fact losing a Pokemon can offer instant momentum; losing one set-up sweeper opens an opportunity for another to set up on whatever just defeated the first one. Usually, players will identify a win condition Pokemon that can beat the opponent given certain circumstances, then will preserve that Pokemon and will play so that the circumstances under which the Pokemon can sweep are met. Hyper offence is effective in basically every tier, as it takes advantage of stall's passivity and balance's dependence on pivoting. Bulky offence is the reverse to hyper offense, and focuses on Pokemon that are strong offensively and have good defensive types that can be used to complement each other. In bulky offence, there's less sacrificing Pokemon, and more switching to preserve Pokemon that constantly threaten the opponent. As such, it's a bit of a mix of balance and standard offence; it focuses on offensive momentum as usual, but also values defensive synergy like balance does.
Another variation of offence is a team based around a Baton Pass chain, which has historically involved Pokemon like Scolipede and Ninjask setting up boosts and passing them on to other Pokemon to sweep through teams. These are generally seen as gimmicky though, and some Smogon metagames have measures in place that prevent the strategy from being overly disruptive.
Stall-based
Stall is an almost exclusively a defensive playstyle. Instead of making predictions and pressuring the opponent offensively, stall opts to block the opponent's strategy by weakening and crippling vital Pokemon on the opponent's team. Its effectiveness depends largely on the metagame, and which Pokemon are currently popular and how well they can handle stall. It's also possible for stall to lose almost purely by simply being matched poorly; if the opponent's team makes it too hard to keep your stall Pokemon around, your stall team likely lacks the offensive backbone to take the win. Conversely, however, stall can match very well against many different teams, and slowly whittle them away until they lose. In the past, entire tiers have been broken by stall teams as they grow too strong, and their repetitive gameplay becomes tiresome -- in such metagames, stall can basically own the ranked ladder until necessary changes or bans are applied.
At its core, stall works by using several defensive "wall" Pokemon that limit the opponent's means of winning the game by spreading status, using entry hazards, phasing set-up sweepers away with moves like Roar and Whirlwind and having the defensive capability to tank attacks from lots of different Pokemon and heal the damage away. (It's as frustrating as it sounds.) Stall teams should be designed almost exclusively with walls, but importantly, with a stall-breaker to go with them. Stall vs. stall matches are very difficult to manage as it's hard for both players to gain any sort of momentum because stall is passive and reactive by design, so having a Pokemon that uses moves like Taunt and a Pokemon to clear entry hazards is very important. The wall Pokemon on the stall team should have good synergy, and collectively have an answer to each common threat in the metagame.
A variant of stall known as "semi-stall" is designed to be less passive and to match better against actual stall teams, but as a result is also less effective against everything else. They make many of the same design choices as standard stall-based teams, but also include set-up sweepers or breakers to finish the match or rid of opposing stall teams.
Others
Most teams generally follow the rough structure as those above, though there are some other variations of these styles that can prove to be effective. As it seems this answer has reached the character limit (woah), I'll include this section as a pastebin link here: http://pastebin.com/GkTnRFg9
This is probably the longest response I've ever written, I think I got carried away. But it should serve as a guide that I'll link back to in the future. Hopefully this answers your question thoroughly. I wanted to cover every aspect of it. Let me know if you have any questions about it!
