Peat has no nutrients within it - its now a scarce commodity, and not commonly used in gardening by amateurs any more. It is, though, a good soil conditioner.

Compost comes in two types - potting and garden - potting compost, as its name implies, is for potting up plants, contains nutrients - multi purpose is an example, and some are loam based and have different strength nutrients within them (John Innes 1, 2 and 3).
Garden compost is well rotted materials for adding to soil in the garden, so this includes organic soil conditioning compost from the garden centre, your own from your compost heap, composted horse manure, leaf mould, etc.

Peat is a single, natural product that forms the base of many of our moorlands. It has been heavily used as a soil conditioner by gardeners for the last fifty years. Unfortunately the mining of it has had a serious effect on the environment of the moors and its use is now being drastically curtailed.
Compost, from the garden center, is a mixture of soil, grit and humus with added nutrients to produce a mix that you can grow plants in directly.