In the mid-1930s, a chap called Tangye-Lean who was an undergraduate at University College
decided to found a literary club.
The original "Inklings" was short-lived, but the habit of getting together to read bits of unfinished works between Tolkien and C.S. Lewis kept on. They, and a few others, gathered in C.S. Lewis' rooms at Magdalen. They decided to keep the name, as it was a rather clever pun, "suggesting people with vague or half-formed intimations and ideas plus those who dabble in ink" (Tolkien "Letters" 388). It is assumed the name came from Tangye-Lean, but it was a common name for literary clubs of the time so there is no particular evidence that T-L or anyone else necessarily came up with it first.
It started mostly with the Lewises and Tolkien along with a few friends at Magdalen. They then took to meeting in various pubs as well. A little later, Charles Williams joined the group as a regular member. Tolkien eventually did not come as often, and the group disbanded as they went their separate ways and the members began to die off as well. The "heyday" might be considered the late 1930s - 1960s.
For full detail about the history and members of the Inklings there is a wonderful site called TheInklings.net that has all the information you could want.