A silver mint comes in a few different forms. The most common mint is a private company. These types of mints are located all around the world and are not typically regulated or run by any governmental body. The other type of mint is a federal or national mint. These are run by the country itself and almost always produce currency items only, with some exceptions. There is a difference between the two because only one has the ability to produce coins and other items with actual value that can be used in everyday monetary transactions.
Private mints can be run by literally anyone. Even you could start up a private mint should you have the ability, equipment, and skills necessary. What these companies do is take bulk silver and form it into refined items. For example, when someone goes into a silver store and sells a lot of cowbells that were made out of silver, they would then be melted. The melted cow bells would then be taken, measured out, reshaped, and produce as brand new pieces of bullion. Silver is extremely reusable and can take many different forms over its lifetime.
If you see a silver bar or silver round that simply has the weight marked on it but no actual country affiliation, you are likely holding a generically produced piece of bullion. The majority of items that private mints create will carry the name of the company in question, the purity of the metal in the piece, and the associated weight. There are some pieces of silver that will not have all or just a few of these markings, but most will meet all of this criteria.
Some examples of private mints include Golden State Mint, Engelhard, Sunshine Minting, Johnson Matthey, Silvertowne, and NTR Metals. There are many, many others as well and it would be nearly impossible to list them all, but these are some of the brands that most silver buyers are familiar with. Engelhard in particular is well-known in silver bullion.
Unfortunately, Engelhard is also the one company in this lot who no longer actively manufactures silver bullion pieces. When they were still operational, however, they produced everything from bars to rounds. They are especially popular due to the great quality that was found in all of their items. These mints each produce bars and rounds, and each one has their logo stamped on each piece.
There is no added value for one brand or another. With clothing you might expect to pay a bit extra if you buy certain brands, but this is not the case with bullion. The mark ups on all general bullion items will tend to hover around the same range. The one time where you will potentially pay extra is if you are buying the items new vs. on the secondary market.
National mints are the most recognizable because they are operated by the countries themselves. The United States Mint, for example, is world renowned. These mints produce a few different types of items. First, your everyday currency will most often be made at one of these mints. This includes everything from paper money to change that you carry in your pocket. Beyond this, and most importantly, national mints create silver coins. Silver coins come in many different forms, from uncirculated to proofs, and they are all made at one of these mints.
Unlike the private mints noted above, these facilities produce monetary items almost exclusively. The only time that a government-run mint would also produce general bullion items such as bars would be if it was for their own personal use or if the mint was simply used by the country vs. owned by it. As you likely already know, items produced at mints will have a much higher premium than those at regular factories or companies. Private mints are very common and easy to find, whereas the mints used by countries for their coins and currency are unique and are much more sparse in population.