Vulgar science graduate, last to apologise, interrupts joke (7)
I believe the answer is:
obscene
'vulgar' is the definition.
(thesaurus)
'science graduate last to apologise interrupts joke' is the wordplay.
'science graduate' becomes 'BSc' (someone with a Bachelor of Science degree).
'last to' says to take the final letters.
'interrupts' means one lot of letters goes inside another.
'joke' becomes 'one' (as in 'that's a good one!').
The last letter of 'apologise' is 'e'.
'bsc'+'e'='bsce'
'bsce' inserted within 'one' is 'OBSCENE'.
(Other definitions for obscene that I've seen before include "One's CBE (anag)" , "coarse!" , "Salacious" , "not decent" , "unsuitable for children?" .)