Formal:
* Arrogant
* Haughty
* Pompous
* Self-important
* Egotistical
* Vainglorious
* Presumptuous
Informal:
* Big-headed
* Stuck-up
* Smug
* Full of oneself
* Snobby
* Narcissistic
More specific:
* Self-absorbed (focused on oneself to the exclusion of others)
* Supercilious (behaving or looking as though one thinks one is superior to others)
* Pretentious (attempting to impress by affecting greater importance, talent, culture, etc., than is actually possessed)
The best word to use will depend on the specific context.