Jestem = Ja jestem = I am
"Jest mi" is completely different thing. It usually describes your state.
"your state" is a definition a little bit too wide, I'd say it describes how you're feeling, both emotionally and the way you feel (perceive) an external stimulus.
In the case of feelings you may usually use 2 ways of expressing it:
Jest mi smutno = Jestem smutny (I feel sad/I'm sad)
Jest mi wesoło = Jestem wesoły (I'm happy/glad)
but here's the twist - as witek7205 said "jest mi + adverb" describes a state of things, very often temporary, while "Jestem + adjective" can have the same meaning or can decribe your permanent characteristics, "Jestem wesoły" can mean "I'm a happy person, I don't worry often"
you can skip the verb entirely, and inverse the order of pronoun/adverb:
Jest mi wesoło = Wesoło mi
jest mi smutno = smutno mi
some of those feelings can be expressed only with "jest me"for example:
Szkoda mi - I regret (can also be translated with "I wish that ...")
Ale mi głupio - I feel so stupid
('ale jestem głupi" means "I'm so stupid", and of course you can say that too, both in Polish and in English, but noone means he's really stupid, as an idiot, you use this sentence in the sense "feeling [temprarily] stupid" for having done some big mistake)
now external stimulus - Jest mi zimno (I'm feeling cold), jest mi ciepło (warm) - you CANNOT say "jestem zimny/ciepły", it's meaningless (of course you may say "ciało jest zimne od 2 godzin" = "the body has been cold for 2 hours")
jestem przykry = i am sad
I'm sad = jestem smutny :)
jestem przykry - I wouldn't say that