What are the requirements or preconditions for someone to ordain? Can a person ordain at any age? Can he or she have children? In a nutshell, what are the things that will not allow someone to ordain?
Disclaimer: This is the current knowledge I have on the Theravadin tradition.
Here is a summary that answers your question:


  Becoming a Novice
  
  The first part of the ordination procedure for bhikkhus is known as
  the Going Forth into Homelessness (pabbajjaa). If it finishes with
  just that — without going on to the Questioning of the candidate and
  the Acceptance of him by all the gathered bhikkhus into the Bhikkhu
  Sa"ngha — the candidate is known as a saama.nera or novice. This is
  usually the case when the candidate is less than the twenty years of
  age necessary to become a bhikkhu. A very young boy is not allowed to
  become a novice either, but the minimum age will vary according to
  place.
  
  A saama.nera wears the 'yellow robe' like a full bhikkhu — except he
  does not have the sa"nghaa.ti (double-thickness robe) — and leads a
  very similar life. In some places a period as a novice forms part of
  the preliminary training to become a bhikkhu, while some men decide to
  remain saama.nera for various reasons. The saama.nera keeps the Ten
  Precepts and the 75 Training Rules (sekhiya) and some other rules of
  the bhikkhu. Later, when he is ready and if he is old enough, he can
  ask the bhikkhu community for full ordination (upasampadaa).
  
  Becoming a Bhikkhu
  
  In the Pali texts, when a man decided to become a bhikkhu, he is often
  quoted as saying: "Confined is the household life, a path of dust; the
  going forth is open and spacious. Not easy is it living in a house to
  lead the religious life absolutely fulfilled and purified, as polished
  as mother-of-pearl. Suppose I were to shave off my hair and beard,
  clothe myself in ochre robes and go forth from homelife into
  homelessness?" (HS ch.19)
  
  However, anyone wishing to become a bhikkhu must fulfill certain
  conditions about which he will be questioned during the actual
  ordination procedure. The candidate must be male and at least twenty
  years old. He must never have committed any grievous crimes and, if
  previously ordained, he must not have been guilty of any Defeater
  (Paaraajika) offences or have entered some other religion without
  disrobing first. He should also be of good
  reputation; fit and healthy enough to carry out the duties of a
  bhikkhu; not in debt; not subject to government service; and have
  permission from parents or guardian.
  
  The Ordination ceremony requires a prescribed boundary (siima), a
  preceptor (upajjhaaya) and a quorum of bhikkhus to validate the formal
  Sangha Act. In the formal procedure the candidate is examined as to
  the necessary qualities and, if all the bhikkhus are satisfied,
  they receive him into the Sangha, the Community of Bhikkhus.
  
  It is in this way that yet another link is added to the
  bhikkhu-lineage. Henceforth, the new bhikkhu can participate (and make
  up the necessary quorum) in future assemblies and help receive other
  new bhikkhus — as bhikkhus have continued to do for two and a half
  thousand years. (See EV,I,p.4; OP)
  
  When a candidate requests full admission to the Community (after
  the saama.nera ordination) he does not make any 'lifetime vows' but
  offers himself for training and instruction under his Preceptor's
  guidance. At the end of the ordination ceremony, the Preceptor will
  immediately instruct the new bhikkhu (or arrange that he is properly
  taught) about the Paa.timokkha Rule and the other principles that all
  bhikkhus should follow and observe.
  
  For the first five years a bhikkhu is called navaka ('new one') and he
  must live 'dependent' (nissaya) on a senior bhikkhu — either his
  preceptor or teacher (aacariya) — training in the ways of a bhikkhu.
  The preceptor and the new monk should be kind and helpful to each
  other, in almost a father-and-son relationship. A new bhikkhu who no
  longer lives under his preceptor must take another senior bhikkhu as
  his teacher and depend on him instead.
  
  For the next five years after his navaka period, the bhikkhu is called
  majjhima, ('one in the middle') and he is allowed to live by himself
  if he is accomplished in certain qualities.
  
  When a bhikkhu has completed ten Rains he is called Thera, which can
  be translated as 'an elder who is worthy of respect.' If he is also
  accomplished in certain extra qualities, he is allowed to give
  ordination as preceptor, to be a teacher, and have young monks live in
  dependence on him.  
  
  -The Bhikkhus' Rules: A Guide for Laypeople  


Note: Requirement of being male is not always necessary and is dependent on the specific monastery in which the ordination takes place.

For a more in-depth answer, see the chapter on Ordination in the Buddhist Monastic Code by Thanissaro Bhikkhu.



Here's a chart of the procedure in some western monasteries (in Asian monasteries the ordination can be immediate):


Some references:  


Ordination at Santi Forest Monastery  
General Information for Potential Anagarikas in UK Monasteries