Facilities at these campsites vary considerably. There are some with next to nothing, some with the basic necessities like running water, suitable for drinking, car parking space and picnic areas, then the really luxurious ones with barbecue areas, showers and toilets. Most have a "recreational area" next to them, the quality of which can also vary a great deal. You are restricted to a maximum stay of seven nights.
Here is a selection of official sites for camping in Tenerife. (On the website linked to above, there are only 13 campsites listed, but, trust me, there are 19 of them.)
Arenas Negras - at Garachico.
Chanajiga - at Los Realejos.
Chio - at Guia de Isora.
El Lagar - near Icod de Los Vinos.
El Contador - at Arico.
Fuente del Llano - at Arico.
Fuente de Pedro - in San Juan de La Rambla.
La Caldera - at La Orotava.
Las Hayas - near Icod de Los Vinos.
Las Lajas - near Vilaflor.
Las Raices - near El Rosario.
Lomo de La Jara - near Tacoronte.
Ramón el Caminero - near La Orotava.
San Jose de Los Llanos - near El Tanque.
Las Calderetas - near La Matanza.
La Tahona - near San Juan de La Rambla.
La Vica - near La Matanza.
Los Frailes - near Arafo.
Los Pedregales - at Buenavista.