All skills are taught by either schools or universities.
There is one type of school and university for each of the seven disciplines: Architecture, Art & Music, Body, Conflict, Leadership, Thought, and Worship. Each teaches a different set of skills. (Actually, there are eight types of University: The University of Progress is an institution of the Stranger.)
Schools teach skills for a cost. All schools of a given type will teach the same skills for the same cost; there is never any need to visit more than one school of a type.
In contrast, universities teach skills for free. However, a given technology must be researched at a university before it will be taught there. People must donate goods to meet the (often very expensive) research cost. Once the research cost is met, the university will begin researching; the new technology will become available one day later. (If you hear that a particular technology is "on timer", this means that the research costs have been paid and people are waiting for the research to complete.)
Technologies at the University of Progress take one week to research, rather than the customary day. Obelisk construction technologies unlock immediately, with no delay.
Often, the first time a new technology is unlocked, new related skills will become available at schools everywhere.
Skills learned from universities may be taught to other players. A person may teach or receive one lesson every ten minutes, and it will takes a number of lessons to teach a skill. (The exact number of lessons depends on the skill.) Neither the teacher nor the student may teach or learn from any other player while teaching a skill, or all progress will be lost and they will have to start over again.
Skills learned from schools may not be taught, with the exception of Forge Tuning.
The map at http://www.atitd.info/ contains a list of what universities have what technologies available.