A reason is a consideration which justifies or explains.
Reasons are what people appeal to when making arguments about what people should do or believe. (Those are reasons in the normative sense.) For example, that a doctor's patient is grimacing is a reason to believe the patient is in pain. That the patient is in pain is a reason for the doctor to do things to alleviate the pain.
In another sense of the term, reasons are explanations of why things happened. (These are reasons in the explanatory sense.) For example, the reason the patient is in pain is that her nerves are sending signals from her tissues to her brain.
A reason, in many cases, is brought up by the question "why?", and answered following the word because. Additionally, words and phrases such as since, due to, as, considering (that), a result (of), and in order to, for example, all serve as explanatory locutions that precede the reason to which they refer.
Normative reasons (AKA justifying reasons) are often said to be "considerations which count in favor" of some state of affairs (this is, at any rate, a common view, notably held by T. M. Scanlon and Derek Parfit).
Explanatory reasons are considerations which serve to explain why things have happened—they are reasons events occur, or why states of affairs are the way they are. In other words, "reason" can also be a synonym for "cause". For example, a reason a car starts is that its ignition is turned. In the context of explaining the actions of beings who act for reasons (i.e., rational agents), these are called motivating reasons—e.g., the reason Bill went to college was to learn; i.e., that he would learn was his motivating reason. At least where a rational agent is acting rationally, her motivating reasons are those considerations which she believes count in favor of her so acting.
Some philosophers (one being John Broome) view normative reasons as the same as "explanations of ought facts". Just as explanatory reasons explain why some descriptive fact obtains (or came to obtain), normative reasons on this view explain why some normative facts obtain, i.e., they explain why some state of affairs ought to come to obtain (e.g., why someone should act or why some event ought to take place).
Philosophers, when discussing reasoning that is influenced by norms, commonly make a distinction between theoretical reason and practical reason. These are capacities that draw on epistemic reasons (matters of fact and of explanation) or practical reasons (reasons for action) respectively. Epistemic reasons (also called theoretical or evidential reasons) are considerations which count in favor of believing some proposition to be true. Practical reasons are considerations which count in favor of some action or the having of some attitude (or at least, count in favor of wanting or trying to bring those actions or attitudes about).