What makes Catch-Rate so flippin' important in Pokémon? Easy, do you want waste all of your Pokéballs on a Rattata? You see, that's why. Catch-Rate is essentially, what it is; the rate at which the Pokémon can be caught. Simple. So with this came with some undefined rules. My weird rules on the matter follows as so:
Now since that's covered, I'm going to go through each rule and check it off, one by one, on my digital clipboard(but not really). Are we cool? Okay, let's do it.
This more or less, doesn't apply that much here. Yeah, kinda contradicted myself there. As Feebas is extremely rare, yet it is has a high catch-rate. But of course, there is legendary Pokémon, who has extremely lower catch rates(excluding Zekrom and Reshiram, who actually have surprising higher catch-rates for Mascot legendaries of all things, even though its still 45, which is relatively low). So in this rule, it more or less applies. Applies with legendaries, but not really Wild Pokémon.
Again, this more or less applies to Legendary Pokémon. Reason being, that legendary Pokémon, has base stat totals, exceeding 600 points. Of course, Pseudo Legendaries, has exactly 600 points in their Base Stat Total, and some other stuff, that I'm too lazy to write out, and its not relevant to the situation.
Basic logic goes as such: Do you really want a super-powerful Pokémon to be extremely accessible? (I guess by logic, ALL Pokémon are extremely powerful)
Do I really, need to explain this? Better that I do. Like I said, the higher a value, the easier it is to catch. Legendaries usually clock in under a 50 catch-rate value, while "first route" Pokémon clock in about a 255 catch-rate value, which is high. 255 is 33.3% to catch at full health, so the number rises with battle conditions, such as damage and status effects.
So depending on your Fakémon, whether it's a legendary or not, will also depend it's catch-rate at the same time. That is all for Catch-Rate.