Poetry that overtly uses the effects of metre, rhyme and form, especially the fixed forms (sonnets, villanelles etc) is known as formal verse. Good examples in the Archive include the extract from Fred D'Aguiar's 'Bloodlines', written in ottava rima, Mimi Khalvati's tight quatrains in 'Don't Ask Me, Love, For That First Love', or Dylan Thomas' 'A Refusal to Mourn the Death, By Fire, Of A Child in London', in which the poet has invented a set of formal restrictions that he adheres to.
Its opposite, strictly, is free verse. Many poets, however, can and do operate in both free and formal ways in their work, and sometimes within the one poem. A classic example of this is T S Eliot's 'The Waste Land', which moves between blank verse and free verse, and shifts in and out of rhyming.