In standard English, you should "keep your pronouns consistent". For example, it would be wrong to say "the children collected HIS prizes". What you ought to say is "the children collected THEIR prizes".
In English, pronouns should agree with number, person and gender. So, for example, "I", "me", "my" and "mine" go together. If someone wrote "I went to the office to collect our books", you would know that the books belong to the speaker and to another person, either someone who has already been mentioned, or to you, as the person being addressed. If someone wrote "She went to the office to collect his books", you would know that the books are not hers. Similarly, someone might ask, "You went to the office to collect my books?" and, again, the speaker and the owner of the books are not the same person. Pronouns thus help us to keep straight who is doing what.
The pronoun "they" is typically considered plural and is thus used to refer to more than one person. You might have learned in fact that this is the only accurate use of the term, but that is not correct. "They", "their" and "them" have been used for a very long time in English when the gender of the person being referred to is unknown. In the more recent past, many writers began using masculine pronouns to refer to unknown persons, followed later by the rather clumsy "he or she", "him or her". The only other gender-free singular pronoun English has is "it", which cannot be used for a person. So, although some people will call this incorrect, it is fine to say or write: "When the customer arrives, give them a refund" if you do not know whether the customer is male or female. This usage is referred to as the "singular they" to emphasise that it still agrees in number with the noun it replaces. Remember that you can sometimes also alter the "unknown" noun to agree with the plural "they": "If any customers come, give them a leaflet". (This wouldn't be a good idea with the refund, however!)
Play this quiz to practise your pronoun skills.