The catch rate determines how easy a Pokemon can be caught.
Usually, the rarer the Pokemon, the harder the Pokemon is to catch. That's not always the case, but that's a general rule. Catch rates are listed by numbers, the highest catch rate is 255, while the lowest is 3.
Different balls have different multipliers.
The normal red & white Pokeball has a multiplier of x1. That means it'll have no extra added effect at catching a Pokemon.
The Ultra Ball has a multiplier of x2, meaning if a Pokemon has a catch rate of 100, the Ultra Ball will make that catch rate 200, causing the Pokemon to be easier to be caught.
The Quick Ball has a multiplier of x5, but only if used on the first turn of battle.
The Masterball has a multiplier of x255, meaning that it will always succeed in catching a Pokemon (as long as that Pokemon is catchable).
So, lets start with an example like Caterpie.
Caterpie has a catch rate of 255. That's the highest catch rate possible, so you can throw any ball at a Caterpie and it will be caught. It's really common, so that's why its catch rate is so high.
Lets move on to a Pokemon like Pelipper. Pelipper has a surprisingly low catch rate of only 45. That means even with an Ultra Ball, its catch rate is only 90. If you use a Quick Ball on a Pelipper on the first turn of battle, it'll have a catch rate of 225.
There are other things in addition to Pokeballs that help catch a Pokemon, such as HP and status conditions.
The lower the HP, the easier the Pokemon is to catch. That's why the move False Swipe is so useful, because it'll always leave the target with 1 HP, meaning you can accurately get a wild Pokemon down to 1 HP to make it easier to catch.
Statuses like freeze and sleep will have a multiplier of x2, while paralyse, poison and burn will have a multiplier of x1.5. Generally, sleep and paralysis moves are the best because freeze is very hard to happen, and burn & poison will decrease the wild Pokemon's HP every turn, meaning it could faint before you catch it.
List of Pokemon by catch rate