During my school days I was a very keen student of mathematics. But circumstances led me to opt for commerce at the college level. Now I wish to continue learning mathematics on a self study basis. First of all I would like to brush up my pre college mathematics (which included basic algebra, number theory, euclidean geometry, basic trigonometry). Can somebody recommend books which explain all elementary concepts, proofs, theorems in brief. I started with What is mathematics but found it to be too rigorous.
It covers the topics you listed above and many more and is written in a way that's meant to prepare one for university/college study.
I would recommend you to try Math Olympiad Books, as they will be very interesting along with providing solid concepts and problems.
You could try Precalculus (Stewart, Redlin, Watson). It barely touches on number theory, and abounds more in exercises than in proofs, but given your context (elementary concepts, not too rigour, self study) it might be useful.
The lecture notes by William Chen cover much of the starting undergraduate curriculum in math, and are very readable.