Firstly, I'd recommend, if you can, get your hands on C.H. Lawrence's Medieval Monasticism. It's the best introduction to the subject and I'm basing most of the answer here on it.

As a general rule, anyone could apply to join a monastery in the central middle ages. In practice however, it was a bit more complicated than that. You had to be literate, in order to take full part in the liturgy and in the prayer and meditation of monastic life. There does appear to have been some leeway here, and we have some evidence that you could spend your private study time improving your reading, but a basic level would have been expected. Secondly, new monks were usually expected to bring an endowment with them to donate to the monastery. This wasn't a strict requirement and, although most endowment was in land, it's certainly possible that other offerings were accepted. As you can expect, these conditions did limit to a large extent those who could become monks. That's not to say you needed to be a noble to become a monk. While there do seem to be monasteries, mainly in Germany, which were pretty snobbish about recruitment, a wealthy peasant could quite easily become a monk.

Peasants didn't need permission to join a monastery, except that of the abbot, who approved all new recruits. Unfree serfs, however, could not become monks, nor could illegitimate children (these rules were common for the clergy as a whole).

Age did play a role in monastic recruitment, particularly the young and the old. Particularly before the 12th century, oblates, boys donated to the monastery, usually around 7 years old, were an important source of monks. This method of recruitment got around the literacy requirement, as the boy would be educated with other oblates within the monastery, although an endowment was still required. There were also a significant number of people who retired to monasteries at the end of their lives, sometimes even becoming monks during their final illness, to make their peace with god and secure an easier route to heaven. Lawrence even gives an example from Canterbury of a man in the 1170s who signed up while his wife was still alive, who is know to us because he arranged for her to be provided for by the monastic kitchen! While there was some criticism of these 'deathbed' monks, they were very common. Most recruits in the prime of their lives would have either been from the wealthier classes, who could afford an education, or from the secular clergy.

There was one other option for a poor peasant who experienced an epiphany however, and that was to become a lay brother, or conversus. Conversi were men who took monastic vows and lived in a monastery, but who were illiterate and took only a limited part in the daily rituals of the monastic office. Their main function was to serve the monastery with their labour, either as farmhands or skilled workers (carpenters, smiths etc.) They appear first at Cluny in the tenth and eleventh centuries, but really take off with the emergence of the Cistercians in the twelfth. (Part of the reason for this is that the Cistercians refused to accept oblates and the minimum age to join was 15. So the illiterate were given another entry route.) The Cistercians' extreme brand of monastic poverty encouraged their conversi to transform the margins of Europe into productive fields for the order and by the 1200s big Cistercian abbeys had hundreds of conversi each, often outnumbering 'choir monks' by 2:1. There was a strict segregation between lay brothers and choir monks, with separate sleeping and dining quarters, which mirrored the social segregation of the outside world from which they were drawn.

One thing I haven't mentioned here is gender, and that's because I know so little about medieval nuns that I don't feel qualified to comment on them. With the exception of the very rare mixed houses, obviously you joined a monastery if you were a man and a nunnery if you were a woman. But beyond that I don't really know how recruitment differed from the standard male model.