A metaphor takes two things and claims they are the same. One way of doing this is by saying X is Y, as in Andrew Motion's 'Veteran'; squeezing two metaphors into the same stanza, he describes a "lilac cloud / which an hour ago was a bird / and is now a shroud." This isn't literally true, but on hearing the lines, it is easy to imagine the cloud firstly taking on the characteristics of a bird, a single small thing moving through the sky, then the characteristics of a shroud, covering us and suggesting deathly omens.
A second, more direct, method is used by Elaine Feinstein, in her poem 'Dad': she asks "What happened, old bull, my loyal / hoarse-voiced warrior?" These lines use the metaphor that her father is a bull, but leap straight to the bull in the knowledge that a reader or listener will make that connection without her needing to point it out; making that connection without showing the steps brings in the emotional response to a great, powerful male presence more immediately.
An extended metaphor takes the original identification and runs with it for a period of time, perhaps the whole poem, such as in Adrian Mitchell's 'Life is a Walk Across a Field' - the poem's title is the metaphor that is extended in the poem. A conceit is a particularly audacious form of extended metaphor. Metaphors may also be implied; when, in 'Catmint Tea', Ciaran Carson says 'Our wild imaginations started to take wing', he is stating an identification between minds and birds.
Some metaphors seem so accurate that they become part of natural speech, and the power of invention behind their original identification becomes inaudible: a flowerbed, or the head of a company, for example. At this point, it becomes a 'dead metaphor' - which is itself a metaphor, in that it makes an identification between a living thing and the turn of phrase.