I want to know what are these abbreviations existed in the picture referring to, and want their names to read about?
" cp, ip, ti, ap, ...etc"
These are the names of standard constituents in X' (X-bar) notation:
CP - Complementizer Phrase ... This particular flavor of X' theory treats the 'fronted' auxiliary as a CP. I don't much care for the name, but the notion that any clause has a 'complementizer' which plays much the same role as a Determiner in a Noun Phrase can be useful.
IP - Inflection Phrase ... This is the 'root' of a predicate, the constituent which bears inflection for tense and agreement.
ti - Trace, co-indexed by the subscript i with the fronted auxiliary ... This is the 'gap' left behind when the auxiliary 'moves' out of the IP into the CP position.
This particular version, with the duplicated trace, is unfamiliar to me. If anybody wants to chip in and explain it, please do!
VP - Verb Phrase ... A constituent composed of a verb and its dependents.
AP - Adjective Phrase ... A constituent composed of an adjective and its dependents.
I have no idea why Mary isn't labeled as an NP (which is sort of like a Noun Phrase except different). Perhaps your source is stripping out labels that don't bear on whatever point is being treated immediately.
The bracket notation is a space-saving way of recording syntactic structure; you can plug it into a tree drawer (without the duplicate ti)and get this:
Scores of books have been written about this approach to syntactic analysis; if it interests you there are very good sites