What is 925 silver?
925 silver is a silver alloy that is 92.5% pure silver by mass, with the remaining 7.5% being made up of other metals, typically copper. 925 silver is also known as "Sterling silver" - and this name derives from medieval times, when sterling was another name for the English or Scottish silver penny.
In those days, a "pound sterling" was literally a "pound weight of sterlings", which since at least prior to 1300 AD were typically manufactured at 925 silver purity.  The term "925 silver" however is a modern one.
Prior to the medieval period, silver coins were not alloyed as they are nowadays. They were made from "fine silver" i.e. pure silver. In modern times, fine silver must be assayed to be verified as at least 99.9% pure silver, also known as 999 silver or "three nines fine". However in ancient times, fine silver was most likely just silver that was as purified as it could be made, with no other metals added.
One of the difficulties with fine silver was that silver is a soft metal and coins or other objects made from it were too easily bent, broken, marked or worn down by ordinary use. Therefore the silver was alloyed in order to increase its strength and functionality. In old times, silver coins' currency value was derived from their actual silver content and so in order for the system to work, silver purity needed to be controlled. A reliable standard was required that met criteria for both functionality and quality. In 1275, King Edward I of England specified that 12 troy ounces of silver for coinage should contain 11 ounces two and a quarter pennyweights of silver.  Then in 1300 Edward I enacted a statute which required that all manufactured silver items must meet the sterling silver standard (925) and must be assayed in this regard by 'guardians of the craft', who used a stamp with a leopard's head to mark approved silver objects.  A pound of 925 silver was created by alloying 222 pennyweights (dwt) of silver (11 troy ounces and two dwt) with 18 dwt of alloy. (222 / 240 = 0.925).
In old times, the "other metal" added to the mix was typically copper, but in modern times metallurgists seek advanced combinations in order to give 925 silver products better properties such as tarnish resistance, strength or casting porosity. 
The Trial of the Pyx
Purity standardization appears to have worked well in England and assaying, the practice of testing silver for purity, appears to be even older than the sterling silver standard. Through previous experience it was known that an unscrupulous moneyer might attempt to pass off silver of lower purity, and the Trial of the Pyx was introduced in the 12th century to test the purity of gold or silver coin being minted.  This trial still continues to this day in England at the Hall of the Worshipful Company of Goldsmiths, being presided over by a judge and an expert jury of assayers. In old times, the trial required that one coin be selected from every ten pounds of silver used for coin.
High standards themselves can cause further problems when met with lower standards: Perhaps as a result of these strict procedures, British medieval silver coin was often of a more reliable quality than that of continental Europe at that time, leading to the export of British silver pennies in large quantities. However, even the English sterling silver coin was, in reality, of slightly varying purity in old times: In a paper printed in the 1874 Journal of the Chemical Society of Great Britain, it was described how the composition of trial plates used in the Trial of the Pyx from the 15th century onwards, varied by a few parts in a thousand.  The paper described how in the alloying process, creating silver trial plates of uniform purity is difficult, owing to the tendency of the silver to concentrate slightly at the center. The paper explains the Trial of the Pyx:
"The composition of the British silver coins does not differ from that of those issued in the time of Edward I, in the annals of whose reign this alloy, which contains 925 parts of fine silver in the 1,000, is alluded to as the "old standard of England."
"The actual manufacture of coins was usually entrusted to a privileged body of men, termed moneyers, who were compelled, under severe penalties, to produce coins of alloys, which were only permitted to vary within certain narrow limits, from the composition of standard trial plates; and the comparison with these plates of pieces set aside from each day's work constituted the sole guarantee to the Sovereign of the purity of the coins, and of the fidelity with which the coinage operations had been conducted.
"Amongst the Cotton manuscripts is preserved the account of a 'trial of the pyx' of gold nobles, in the 23rd year of Edward III (1350) ; the coins were to be compared with one ounce of florins of Florence, kept in the Treasury for standards, and, according to Ruding, the first mention of trial plates occurs in records of the 17th year of Edward IV (1477).... although the standards of fineness were always prescribed by law, these plates nevertheless have at times been very inaccurate." 
925 Silver in Modern Use
925 silver was used for British silver coinage from the 13th century through to 1920 (when it was replaced by 'cheaper' .500 silver). Since 1947, British coinage has been made from cupro-nickel alloy and contains no silver.
Britannia silver (958) was introduced as a higher purity standard for silver plate in 1697. This was done in order to limit the illegal practice of coin clipping, as shavings obtained from 925 silver coins would no longer be of sufficient purity to melt down and use for silver plate. 
Later, when coin clipping was no longer practiced, sterling silver returned to being the principal standard used for silver household objects, status symbols which reached the height of their popularity in the Victorian era. In Victorian times, all kinds of household objects from perfume bottles and cigarette cases to mirror frames and of course cutlery were made from 925 sterling silver. In the modern day, 925 silver continues to be one of the most common standards used by silversmiths for silver jewelry, flatware and other silver objects.
Often when we see an item made of silver (or sometimes other precious metals), there will be a stamp on the metal somewhere with a three figure number. This stamp may be tiny and may require either very good eyesight or a magnifying glass to identify. This three figure number is a system called millesimal fineness and is used to provide a visual indication of the purity of gold, silver or platinum. 925 is a common example of millesimal fineness, very often seen on silver jewelry - and this of course means the silver item should be .925 pure silver, or 92.5%.
To calculate the amount of actual silver in a 925 silver item, first weigh the item and then multiply the weight by 0.925 .
Other millesimal fineness numbers sometimes seen include 958 ("Britannia silver"), 999 ("Fine silver" or "Three Nines Fine"), 950 ("French First Standard"), 900 (USA coin silver), 833, 995 and several others. Gold is often seen stamped at 916, which is equivalent to 22 carat, 585 (14 carat) or 375 (9 carat). Platinum jewelry is most often seen at 950 purity. 
Does a "925" stamp guarantee silver purity?
Just because a 925 stamp appears on an object, it does not, strictly speaking, guarantee that it is actually 925 silver - although it is illegal to pass off silver of lower purity as 925.
It is very important to understand that the '925' stamp or the 'maker's mark', appearing by themselves, are not hallmarks, although they are very often miscategorized as such because a genuine hallmark often includes a millesimal fineness mark. A 925 mark may be assigned to a product by its maker; whereas a hallmark is a specialized mark that can only legally be assigned by an authorized assay office. A hallmark is an actual guarantee by authority that the precious metal item has passed a formal, authorized assay test of purity.
In Great Britain, most silver jewellery is hallmarked, which means it has been assayed and its purity is guaranteed.
925 Silver - Info Sources
 Oxford English Dictionary, "Sterling"
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