Formal:
* Arrogant: Implies a feeling of superiority and a disdain for others.
* Haughty: Similar to arrogant, but with a sense of aloofness and disdain.
* Pompous: Marked by excessive self-importance and a tendency to show off.
* Pretentious: Trying to impress by affecting greater importance, talent, culture, etc., than is actually possessed.
* Self-important: Having an exaggerated sense of one's own significance.
* Supercilious: Behaving or looking as though one thinks one is superior to others.
Informal:
* Big-headed: A simple and direct way to say someone is conceited.
* Egotistical: Focused on oneself and one's own needs and desires.
* Narcissistic: Having an excessive interest in or admiration of oneself.
* Smug: Having or showing an excessively self-satisfied feeling or air.
* Vain: Having an excessively high opinion of one's appearance, abilities, or worth.
Other options:
* Boastful: Tending to brag about oneself or one's achievements.
* Self-absorbed: Preoccupied with oneself and one's own concerns.
* Overconfident: Having an excessive belief in one's abilities.
* Self-righteous: Believing oneself to be morally superior to others.
The best adjective to use will depend on the specific context and the tone you want to convey.