"Postcolonialism" and "Post-colonialism" are the same thing. The most common accepted form of the word is that without the hyphen. "Postcolonialism" is a field of study that incorporates many sub-fields, including history, anthropology, literature, and what is known as "area studies," specialization on one or more regions of the world (e.g., the Middle East or South Asia) or specific countries (e.g., Egypt, Russia, France, etc.). It examines the effects on native or indigenous cultures of outside (usually referencing European) influences imposed through the process of invasion, occupation, and exploitation (i.e., colonialism). The formal structure of colonialism having been removed through the process, often violent, sometimes peaceful, of decolonization, what is left behind is a complex amalgamation of foreign and domestic influences. There is no shortage of good examples, as colonization affected most of the world, either from the perspective of the imperial power or from that of the occupied nation. Among those examples is the enduring influence of French colonialism in Southeast Asia, where the French language remains in use decades after independence was attained in such nations as Cambodia, Laos and Vietnam (although, each of these nations uses as their primary language that which is native to the ethnicities involved) and French architectural influences dominate many towns and cities. The effects of colonialism, however, run much deeper in terms of the watering-down of indigenous influences in favor of those that were imposed by the outside. Similarly, the use of English across much of India is an enduring legacy of that country's many years of incorporation into the British Empire, as are the legal and civil service systems that are dominant in the ancient culture of India.
In short, "postcolonialism" refers to the long-term influences of foreign occupation on countries and regions formerly colonized by wealthier, stronger European powers. It is not technically incorrect to hyphenate the phrase, but the non-hyphenated form is acceptable.
There is no difference between "post-colonialism" and "postcolonialism". They are simply two ways of spelling the same thing. The more common usage is "postcolonialism" but some spell-checking software will flag this as an incorrect spelling but not flag the hyphenated version as incorrectly spelled.
Postcolonialism is generally a body of literary and cultural theory which studies the cultural impact of imperialism especially on African, Asian, and South American nations. It tends to investigate both the clash of cultures involved in the colonial experience and the process of decolonization. It studies ways in which colonized nations simultaneously assimilate to and appropriate the cultures of imperial powers while subverting them, and is often focused on points of resistance. Postcolonial theorists are also interested in the intersections of race, gender, and colonialism as processes of oppression.