What is the meaning of "size" of word2vec vectors [gensim library]?
It is not the case that "[word2vec] aims to represent each word in the dictionary by a vector where each element represents the similarity of that word with the remaining words in the dictionary". 
Rather, given a certain target dimensionality, like say 100, the Word2Vec algorithm gradually trains word-vectors of 100-dimensions to be better and better at its training task, which is predicting nearby words. 
This iterative process tends to force words that are related to be "near" each other, in rough proportion to their similarity - and even further the various "directions" in this 100-dimensional space often tend to match with human-perceivable semantic categories. So, the famous "wv(king) - wv(man) + wv(woman) ~= wv(queen)" example often works because "maleness/femaleness" and "royalty" are vaguely consistent regions/directions in the space. 
The individual dimensions, alone, don't mean anything. The training process includes randomness, and over time just does "whatever works". The meaningful directions are not perfectly aligned with dimension axes, but angled through all the dimensions. (That is, you're not going to find that a v[77] is a gender-like dimension. Rather, if you took dozens of alternate male-like and female-like word pairs, and averaged all their differences, you might find some 100-dimensional vector-dimension that is suggestive of the gender direction.)
You can pick any 'size' you want, but 100-400 are common values when you have enough training data.
Assume that we have 1000 words (A1, A2,..., A1000) in a dictionary. As fa as I understand, in words embedding or word2vec method, it aims to represent each word in the dictionary by a vector where each element represents the similarity of that word with the remaining words in the dictionary. Is it correct to say there should be 999 dimensions in each vector, or the size of each word2vec vector should be 999?
But with Gensim Python, we can modify the value of "size" parameter for Word2vec, let's say size = 100 in this case. So what does "size=100" mean? If we extract the output vector of A1, denoted (x1,x2,...,x100), what do x1,x2,...,x100 represent in this case?