The suffix '-chester' is often found in place names in Great Britain, and through imitation, America. It has its roots in the Latin word castra, meaning a military camp or fort. This was adapted by the Saxons in the form ceaster, and when they invaded England it was used to name locations that had been Roman encampments, or in some cases, pre-Roman forts.
This suffix appears in various other forms: in England -caster (e.g. Lancaster and Tadcaster) and -cester (e.g. Gloucester and Leicester) are commonly used. When the form -cester is used, it is usually pronounced as "-ster", which Americans are likely familiar with through the orthographic disaster that is Worcestershire sauce. It does not appear only as a suffix; not only do we have the city of Chester, but the place names Caister (e.g. Caister-on-Sea), Caistor (e.g. Caistor Saint Edmund), and Cheshire also come from Saxon ceaster.
Most -chesters appear in England and its colonies, but in Wales the form Caer or Car is used, usually as a prefix (e.g. Caerphilly or Cardiff). Ireland, not having been conquered by the Saxons, does not use any form of ceaster; the Irish towns ending in -ster get their names from the Old Norse staðr, meaning 'land'. As far as I can find Germany, including Saxony, have completely dropped -chester in favor of -burg, which is German for 'fort' or 'castle'.
Most English '-chester' towns have names that are a mix of Saxon and Celtic languages. For example, Manchester comes from the original Celtic name Mamucio + -chester, Gloucester from Glevum + -cester, and Doncaster from Danum + -caster. There are exceptions, of course: Rochester is pure Old English, originally Hrofae + casetre, meaning 'roof fort'. America has gleefully added to the mix in a linguistically inane frenzy, creating names such as Westchester (and West Chester), Chesterville, and Chester Borough, to name just a few.