'Vet' is 'know' as in 'I know'. 'Kjenner' is the verb 'to know' as in 'I know him' 'Kan' means 'able to' or 'can' as in 'I can do that'.

Å vite = to know, to be aware of, to be certain, to recognize (in some contexts, interchangeable with "å skjønne" = to understand) -jeg vet svaret = I know the answer -vet/skjønner du hva du skal gjøre? = do you know/understand what you're supposed to do? -jeg vet hvem han er = I know who he is Å kunne = to be able to/manage, can, to know, may (in some contexts, interchangeable with "å klare" = to manage/be able to) -kan jeg komme? = may I come? -jeg kan å lese = I can read / I know how to read -hvis hadde kunnet hopp -jeg kan ingenting = I know nothing (jeg vet ingenting has the same translation but the meaning behind it may differ from context) -jeg klarer ingenting = I'm not able to do anything -jeg kan det utenat = I know it by heart Å kjenne = to know, to sense/feel/notice (in some contexts, it's interchangeable with "å føle" = to feel) -jeg føler/kjenner meg kvalm = I feel nauseous -jeg kjenner ham = I know him (as in an acquaintance [en bekjent]) -jeg kjente en rar lukt = I sensed/felt [smelled] a weird smell