Please note that I am answering this question from a U.S. perspective.
Handicapped and disabled are common and accepted in everyday use. We have handicapped parking spots and disabled veterans, and most people do not take offense to those two terms.
Special needs is also acceptable, although many people use it to specifically refer to learning disabilities, since the term is often used in the context of schools.
The terms invalid, incapable, and cripple[d] are considered archaic and offensive to most people, since they emphasize that disabled people cannot do certain things.
There are people who support alternative terms like differently abled, but I suggest that you avoid using them, since the ambiguity of these words can lead to confusion. This term is only really used in America, and according to the Oxford Dictionaries Online, it "has gained little currency, … and can seem overeuphemistic and condescending. The accepted term in general use is still disabled."
The above examples are used when talking about people with disabilities in general, no matter what condition they may be suffering from. Terms like monoplegic describe a specific condition with a narrow medical definition, so you should only use them when you want to talk about that condition specifically.
How to use them
Several major academic and medical organizations (like, for example, the American Psychological Association) suggest that when writing and talking about people with disabilities, we should talk about the person first, and then the disability. That is, we should say a man with a mental handicap instead of a mentally handicapped man. This is to avoid dehumanization of the person, and is known as people-first language.
Hope this helps.