Word embedding is a representation of a word, usually with a vector of values.
The simplest way to represent words is by using one-hot encoding - i-th word in the vocabulary will have a vector with 1 in i-th position and 0 in all the others. But this isn’t efficient: such vectors are too sparse, too long and provide too little information.
Slightly better approaches are count-based encoding and TF-IDF.
The first paper suggesting using embeddings was A Neural Probabilistic Language Model (2003) - they authors trained a basic language model to learn a distributed representation for words. Another early paper was A unified architecture for natural language processing: deep neural networks with multitask learning (2008), where the words were “embedded into a d-dimensional space using a lookup table” - basically, a first modern mention of word embeddings.
But the Word2Vec paper Efficient Estimation of Word Representations in Vector Space was the start of the wide-spread use of word embeddings. Word2Vec provided an efficient way to train shallow neural networks on large corpora to produce high-dimensional vectors, such that words appearing in similar contexts have similar vectors.
GloVe GloVe: Global Vectors for Word Representation used global word co-occurrence statistics to produce embeddings, using a different approach (leveraging matrix factorization and corpus-wide statistics)
fastText Enriching Word Vectors with Subword Information extended Word2Vec by representing each word as a bag of character n-grams, enabling the model to generate vectors for rare or misspelled words by composing subword information
Later, BERT-like models started being used for word embeddings. ELMo produces contextual embeddings.