It is important to realize or remember that coin quality is not a precise science. Opinions about quality will vary between countries, mints and individual collectors. Even expert dealers will grade the quality of a coin differently on different days. As there are no worldwide standards to go by, individual mints can refer to their qualities by whatever name they want. It is therefore the case that some mint's brilliant uncirculated quality is better than other mint's proof quality, for example. However, these are the most commonly used quality standards and their definitions.
An abbreviation for "uncirculated". Defines a coin struck by a high-speed press (700+ coins per minute), where the coins end up in a hopper together with other coins. These coins may have surface marks, scratches or light abrasions from the minting process, but have not yet been packaged in rolls/bags and placed into circulation through commercial banks. In new condition as issued by the mint, but not necessarily perfect. Details are supposed to be sharp and the luster approaching the state of the coin at the mint, prior to general circulation
An abbreviation for "fleur-de-coin" (flower of the die). Similar to an uncirculated coin, but without marks from the minting process as FDC coins are largely prevented from touching other coins during manufacturing. Very similar to, or sometimes identical with, brilliant uncirculated or specimen coins (see below). Not used much by world mints anymore.
An abbreviation for "brilliant uncirculated". Most mints define these coins as uncirculated coins (see above) that have been washed and specially treated before being placed in sets. Most of the time the minting dies and coin blanks get better polished and treated before the minting process starts. In addition, most mints see to it that these coins actually don't have the marks from the minting process that regular uncirculated coins would have. Some mints, like denmark, norway and portugal, actually mint their brilliant uncirculated coins one-by-one in a proof press using specially prepared uncirculated coin dies, sometimes with more than one strike. Most brilliant uncirculated coins have an appearance closer to uncirculated than proof quality. Sometimes described as "superior normal strike".
Specimen is used to describe any coin produced to a particularly high standard of finish. Used frequently before the 1970s, most of these coins would now be called proof quality. In recent years, some mints have produced especially good versions of its uncirculated coins which they call specimens. These are ordinary uncirculated coins which have been handled individually and with greater than normal care, to avoid most, but not all, of the surface blemishes which occur due to bulk handling. In that regard they would be very similar to brilliant uncirculated or FDC coins.
Typically, these are coins that have been struck on proof quality coin blanks, using brilliant uncirculated minting dies. This leaves the coin with a very reflective surface due to the brush polished blanks, but don't have the characteristics of polished proof coins with mirrorlike surfaces and frosted reliefs because the dies are not proof quality. The process makes a beautiful coin at an affordable price.
Most mints have adopted the british standard where the coins have a mirror surface and frosted reliefs. The blanks are typically polished with brushes before minting to remove residue from soap and polish paste that can interfere with the end result. The dies used in this process have sharper detail and do not last as long as uncirculated dies. In addition, the coins are minted using more tonnage of pressure, so that the metal will flow more and require a one-by-one minting technique. Often, 2-3 strikes are utilized, depending on the hardness of the metal. After each strike the dies are inspected and often re-polished by hand. Platinum can require as many as 10-12 punches before a coin is finished. Each additional strike increases the discard rate for the manufacturer, and increases the price of the finished coin. Proof coins can be expected to be absolutely perfect, without blemishes, when looked at with the naked eye. Some mints, like the norwegian, define its proof quality not according to the amount of mirror surface and frosted relief, but according to the sharpness in the detail of the finished coin. This moves the quality towards describing a process instead of a finished product.