There are several types of vector representations for words and characters, I'm assuming here the primary interest is dense representations that are used commonly in deep learning today.
First, Some Background
The purpose of word or character vectorization was to transform sentences or documents into feature sets that could be used for machine learning. In traditional NLP, several methods were used for vectorization such as count vectorization and TF-IDF vectorization. These methods essentially would produce matrices based on the occurrence of words in documents. For instance, count vectorization of a set of documents would return a matrix with size
n_documents X n_words (rows of documents, columns of wrods). Each column would be a vector representation of a word. Considering that there are a huge number of words in languages and only a few words would appear frequently, these vectorization matrices would be enormous, but contain mostly zeros i.e. sparse representations. There are a few problems with this approach:
- Extremely high dimensional data doesn't work well in most ML methods i.e. curse of dimensionality.
- These sparse representations are inefficient (particularly memory inefficient). Often needs to be compressed by some method such as PCA and data reduction such as removal of stop words.
- These vectorization methods are dependent on the training samples and doesn't generalize well.
Because of these problems, researchers were motivated to find better representations.
Dense Representations of Words
The two seminal works on dense representations were Word2Vec and GloVe. These methods produced dense, fixed-length vector representations of words rather than being dependent on the number of training samples, which solved problems 1 and 2 (from above). To solve problem 3, these models were trained on very, very large corpora (see section 4.2. of GloVe) .
The result from these methods (which was a surprising results) was that the dense vectors appear to contain some amount of semantic information i.e. in some sense word meaning was embedded into these vectors. Consider the following quote from the Word2Vec paper:
"Somewhat surprisingly, these questions can be answered by performing simple algebraic operations
with the vector representation of words. To find a word that is similar to small in the same sense as
biggest is similar to big, we can simply compute vector X = vector(”biggest”)−vector(”big”) +
vector(”small”)... When the word vectors are well trained, it is possible to find the correct answer (word
smallest) using this method." 
To answer the first question (what are these representations and why are vector used):
These dense word vectors efficiently represent (in some sense) words in terms of semantics.
Ok so where does the vector length come from? For Word2Vec and GloVe, typically a size of 300 is used/recommended. This comes from the GloVe paper. The following image  from the GloVe paper shows the performance of GloVe vectors in semantic and syntactic tasks. The performance essentially flattens out at a vector size of 300.
These dense representations has nice properties and have some significant advantages over traditional methods, but they still require a large embedding matrix. The GloVe paper mentions that only the 400,000 most common words out of several billion were used. So scale is still an issue. Also, when only a small subset of words are selected for embedding a large number may be "outside the vocabulary" (OOV). These words all get mapped to a single OOV vector. Additionally, there are other issues like misspellings. This motivated embeddings of characters and subwords.
As deep learning in NLP exploded, larger and larger vocabulary sizes where needed. Character and subword embeddings were an attempt to limit the size of embedding matrices such as in BERT. However, these types of embeddings do not encode the same deep sematics that word embeddings encode.
Character embeddings are constructed in similar fashion to the way that word embeddings are constructed. However, instead of embedding at the word level, the vectors represent each character in a language. For example, instead a vector for "king", there would be a separate vector for each of the letters: "k", "i", "n", and "g". As mentioned these types of embeddings do not encode the same type of information that word embeddings contain. Instead, character level embedding can be thought of encoded lexical information and may be used to enhance or enrich word level emebddings (see Enriching Word Vectors with Subword Information). While some research on use of character embeddings has been done (see ), character level embeddings are generally shallow in meaning.
As mentioned in  character-level embeddings have some advantages over word level embeddings such as
- Able to handle new slang words and misspellings
- The required embedding matrix is much smaller than what is required for word level embeddings.
 Efficient Estimation of Word Representations in
 GloVe: Global Vectors for Word Representation
 Character-level Convolutional Networks for Text Classification