What is a patent?

There is a common misconception that a patent is "A government grant that gives the holder exclusive rights to a process, design, plant or new invention for a designated period of time."
In fact, a patent is not an exclusive right to anything at all, but rather the right to exclude OTHERS from

• making,
• copying
• using,
• selling (or offering to sell), or
• importing

any infringing articles or processes, from the time the patent is published to the time it lapses.

In other words, just because you may have a patent, it does not give you any particular right to carry out your own invention, especially if portions of your invention were patented by others.
The public policy underlying patents is that inventors should have the opportunity to benefit from their inventions, to the exclusion of others or by licensing to others, in exchange for publishing a full description of the invention and how to make it. That way, inventive knowledge is circulated freely, but cannot be exploited directly by others without risking penalties. To capitalize freely upon someone else's patented invention, you would need to find a different way to do the same thing (or wait 20 years, or go overseas), but at least the seed for ingenuity is planted.
The most common patent is the "utility" patent, which, in the U.S., is effective for 20 years from the date of filing, unless it was filed prior to 1995, in which case it is effective for 17 years from date of issue, or 20 years from date of application, whichever is later. A "design" patent merely protects a new, original ornamental shape and appearance of a specific class of objects (such as shoe treads, perfume bottles, household appliances), and only for a limited time of 14 years from date of grant. A "plant" patent covers a new and distinct, asexually reproduced species of plant, for the same period of years as for a utility patent (i.e., 17 or 20 years). Some types of patents require periodic payments after they are issued, or they lapse and become unenforceable.
Each patent application is examined to determine if it covers statutory material (allowed to be patented), which is "useful", new with respect to prior disclosures, and not obvious to one of ordinary skill in the art, provided with all of the relevant prior disclosures.
A "utility" patent covers any "new and useful process, machine, manufacture or composition of matter, or any new and useful improvements thereof." This has been interpreted in U.S. courts to include inventions that incorporate computer programs (software), as well as "methods of doing business," although these are each controversial in international relations.
To obtain patent protection in foreign countries, a patent application must be filed in each country, according to the terms of their national laws. International rules serve to streamline this expensive process by permitting applicants to consolidate examinations and modifications in a single process prior to invoking the application process in each desired nation.
Patent applications can be difficult to properly draft and usually result in administrative arguments with the patent examiners that can take (literally) years to complete, and there is no guarantee that a patent will ever be issued from a given application. If you have a valuable invention, you should seek legal counsel of a registered patent attorney to determine when, where, how and why you should file for a patent.  

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There are many questions related to patents. You may wonder
"What is patent pending" on a device you purchased, or "How do you obtain a patent?", or "Are patents valuable?" or even, "What do I do if sued for patent infringement?"

In general, patent answers can be complicated by which country you're concerned with, how old the patent is, and so forth. However, you may find the answers to the related questions helpful. Otherwise, feel free to ask a specific question about patents and someone will try to help you.
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ADDED: For SHORT and CONCISE answer see the discussion page.