The most common and most likely answer to this crossword clue is the 8 letter word ARROGANT
I've seen this crossword clue in The Times
A
R
R
O
G
A
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T
More clues leading to the result ARROGANT
- Superior unqualified to cover leave, on reflection
- Conceited and brazen, holding progress up
- . Proud Agra, torn apart
- Conceited Arab with bombastic speech about origins of oil glut
- Superior skill overcomes Josh’s predecessor
- Disdainful, a bishop over tango dancing
- Overweening republican seen in old marketplace returning books
- High and mighty go back, breaking rank
- Superior total, sealing game up
- Lofty music perhaps including organ playing
Buddy explains!

The straight part
We know that "Conceited" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case conceited = arrogant.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ARROGANT mean?
ARROGANT adjective- Having excessive pride in oneself, often with contempt or disrespect for others.