A core is a set of two or three Pokemon in a team that either work very well together or cover many of each other weaknesses creating a sturdy group of Pokemon that can be challenging to take down if the person uses a good sense of prediction.
Offensive cores usually involve a group of Pokemon that support each other well in terms to weaknesses and ability to switch in a lot. Good offensive cores include Scizor + Rotom Wash, a core that works because of the two Pokemon's ability to switch in on thing and take hit repeatedly and can use U-Turn and Volt Switch to do so efficiently. Jirachi + Machamp is also a common one which relies mostly on hax to win. Jirachi uses Thunder Wave or Body Slam to paralyse the opponent, and then Machamp comes in and confuses it with DynamicPunch (which has infinity accuracy thanks to No Guard). Machamp will also move first thanks to the speed drop provided by paralysis.
Defensive cores are more technical however. They involve a group of Pokemon that can cover each other's weaknesses, wall most of the tier and counter many different teams such as stall and hyper offence.
The most well known defensive core is SkarmBliss. This core is classic and has existed since Gen 2 and still remains just as effective now as it was back in the stall generation. Blissey walls every special attack that Skarmory may be frightened of and Skarmory does the same for physical attacks aimed at Blissey. They can both heal and Blissey can heal Skarmory with Wish and thanks to Sturdy and Blissey's ridiculous HP, Skarmory just won't die. Just don't let this core set up on you, because you'll have many problems if you do.
Smogon lists some more cores here and explains them in more detail.
Synergy just means a team as a whole with the ability to cover each member's weaknesses and also have an answer to almost every Pokemon in the game. The less weaknesses there are and the more Pokemon you have that cover each type, the better the team works together. So yeah, you'd have bad synergy is you had a team of just Water Pokemon for example since there'd be nothing that can deal with Electric and Grass attacks. However, if you add a few Pokemon that can cover these weaknesses and another to cover the new weaknesses, the team would work better. So if I added Ferrothorn to the pure Water team to take Electric and Grass hits and Heatran to turn Ferrothorn's Fire weakness into a boost, the team would have better synergy, though the team would still not be ideal since if Ferrothorn goes down you are in trouble. This is why having many Pokemon able to cover a weakness is so important.