When to use いる/ある　vs　います/あります

I frequently come across this while doing Sentence Complete excercises. Some sentences use ～ある while some use ~あります. I know it's just a conjugation, but I don't know WHY it is or isn't conjugated. Some examples I can remember are:

かのじょはとても思いやりがあるね？ - She's very considerate, isnt she?

その机の上に猫があります。- There's a cat on top of that table.

Why isn't the first ある conjugated, but the second one is? I've been assuming it's like だ vs です, where they mean the same thing, but its more polite to use the conjugation instead of the plain form, but if that's wrong, I'd really like to know.

This is mainly due to formal vs casual speech.

When talking with friends, you would use casual speech, meaning you don't need to conjugate the verb. However, if you're talking to someone like a professor, you would use formal speech, where you conjugate the verb. I hope this helps.