A plimsoll is a type of athletic footwear constructed from a canvas upper and rubber sole. Its development is seen as the origins of sports footwear, and it was originally development as beachwear by the Liverpool Rubber Company (UK). Originally known as sandshoes, they only became known as plimsolls in the 1870s. This name probably originated from the colored horizontal band that joins the upper to the sole that resembles the Plimsoll line on a ship's hull.
In many parts of North America, plimsolls are referred to as "Cons". This refers to a popular brand of Plimsolls - Converse's Chuck Taylor All Star shoe.
general word for slip-on shoes you wear in the house is just slippers. That word covers every type, from leather slides to snoozies. As you say, you can call them all slippers, but if you want to be more precise, t...hen you can say slides etc. Sneakers is a difficult one because in American English it can be used for both canvas plimsoll/tennis shoes, and also for typical Nike, Adidas etc. sports shoes which in the UK we would call trainers. Is that clear? Yes plimsolls and tennis shoes are synonyms (see the Wikipedia page I linked to earlier) and no you don't only use tennis shoes for tennis! I hope that helps. Climbing Shoe (kletterschuh)