With stamp forgeries, there are a number of terms that are used interchangeably by many collectors, but the terms actually refer to very different types of things. The common philatelic terms are as follows:
Counterfeit (postal forgery) - these counterfeits were created for a different market than philatelists. Such postage stamps are usually common definitive postage stamps that are counterfeited to defraud a particular government's postal service.
Some postal forgeries are actually quite collectible, and can be many times more expensive than the actual postage stamp they were originally made to imitate. Postal stamp forgeries have been around since the beginning of postage stamps, with counterfeit postage stamps showing up in the mails, not too long after the issue of the Penny Black. This was the primary reason that many countries experimented with watermarks, special papers, soluble inks, secret marks, silk threads, etc., in order to prevent counterfeiting.
There are also "political postal forgeries", as well. During World War II, the Axis and Allied governments occasionally printed imitations of the definitive postage stamps of their enemies, usually with insulting pictures or slogans on them. They were never circulated in the actual countries, but they are popular with stamp collectors today, and some of them are quite valuable.
Forgery - postage stamps produced to defraud collectors and to defraud stamp-issuing governments. Knowledge is an important tool in detecting forgeries. A person that can easily spot stamp forgeries can usually save a lot of money in expertizing fees. All serious specialists in 19th Century classical material should study forgeries and have reference copies of them, along with the genuine stamps, in their collections.
Fake - the alteration of a genuine stamp to make it appear as something else. Fakes might refer to cancels, overprints, added or clipped perforations, design alterations, etc.
Fake cancels are a real problem for many postmark collectors, and they can be difficult to detect.
Some very rare early U.S. coil and imperforate stamps have also been faked, by trimming the perforations off of an inexpensive sheet-stamp with the same appearance.
Official Reprints - postage stamps that are no longer valid for postage, but that are produced by governments, to meet a philatelic need.
The People's Republic of China did this, after supplies of their 1949 first issues were exhausted, in order to provide the stamps for collectors.
Most of the German States, years after the postage stamps had become obsolete, made reprints of their first issues, for sale to collectors. Many of these reprints, or "nachdrucken", in German, are quite valuable now, and they are collected alongside the original government issues by specialists. They usually have printing on the back of them, either "ND" or "Nachdruck", or they have slightly different characteristics than the original postage stamp issues.
In 1875, the U.S. government made reprints of every postage stamp that had been issued since 1847, including the then current definitive series. These were created to provide stamp collectors with mint examples of the early U.S. postage stamps. They were sold, by mail-order and at face value by the U.S. Post Office Department. Oddly, not many of the reprints were sold, so today they are all worth a fortune -- many times the mint catalog values of the original postage stamp issues!