In fortress mode, a peasant is basically an unskilled laborer, a dwarf who is at most of dabbling skill when it comes to labor. Peasants are automatically assigned only to hauling tasks, and may be important in this regard - a large body of haulers will keep your dwarves from dealing with the tedium themselves, allowing them to go about their work more efficiently. They appear teal on the unit list.
You get peasants in two ways. The first is when children reach their twelfth birthday, if they do not meet the parameters for skill assignment, they automatically become peasants. Children that have been around a while generally gain levels in the grower skill, because "Harvest Plants" can be done by any dwarf, children included, unless you have changed this to farmers only in the standing orders.
The second way is to get them in migrant waves. Note that military and social skills are not considered when labeling a dwarf - a peasant may well be a skilled military fighter, but still only haul around stones at the workplace. More often than not, peasants in migrant waves have some military skills and make good drafting targets for your military. Note that peasants in the military that are sent off-duty have an additional negative thought from being relieved of duty; this can be solved by assigning them to a easily-trained or needed task (engraving, for instance) until they reach at least novice skill at it.
In many fortresses, peasants are forced to remain unskilled laborers by only assigning them hauling tasks. Peasants can be used to exploit the mood system. By assigning a peasant a skill which requires significant cost to the fort to train (i.e. any of the smithing skills, or glassmaking or pottery if there are no domestic sources) until he reaches a level of novice, then un-assigning the skill, the next time a stray mood is assigned the likelihood of a legendary laborer in the critical field is heightened. Due to skill rust, peasants assigned in this way to the "mood lottery" can be reassigned easily later on.