There were three different sets of events held in the sport we now call tennis. In 1908, "tennis" usually referred to the specific indoor game often called court tennis, real tennis, royal tennis, or jeu de paume, and which was contested at the 1908 Olympics.
In those days, "tennis," as we now know it, was usually contested on grass courts, and was termed lawn tennis.
However, in addition, an indoor version of the sport was also on the 1908 Olympic program, termed covered court tennis.
The lawn tennis events of the 1908 Olympics were held at that renowned facility, the All-England Lawn Tennis & Croquet Club, better known to the rest of the world as Wimbledon.
The covered court events were held at Queen's Club in West Kensington, which also hosted the jeu de paume and racquets events of the 1908 Olympics.
The 1908 Olympic events were basically European and British Empire championships. No Americans or Australian tennis players competed in London.
Outside of Europe, the only other players were from Canada and South Africa.
In the covered court events, the representation was even more limited; only tennis players from Great Britain and Sweden took part.