What are the differences between vector, matrix and array data types?

There is no difference between a matrix and a 2D array:

> x <- matrix(1:10, 2)> y <- array(1:10, c(2, 5))> identical(x, y)[1] TRUE...

matrix is just a more convenient constructor, and there are many functions and methods that only accept 2D arrays (a.k.a. matrices).

Internally, arrays are just vectors with a dimension attribute:

...> attributes(x)$dim[1] 2 5> dim(x) <- NULL> x [1]  1  2  3  4  5  6  7  8  9 10> z <- 1:10> dim(z) <- c(2, 5)> is.matrix(z)[1] TRUE

To cite the language definition:

Matrices and arrays are simply vectors with the attribute dim and optionally dimnames attached to the vector.

[...]

The dim attribute is used to implement arrays. The content of the array is stored in a vector in column-major order and the dim attribute is a vector of integers specifying the respective extents of the array. R ensures that the length of the vector is the product of the lengths of the dimensions. The length of one or more dimensions may be zero.

A vector is not the same as a one-dimensional array since the latter has a dim attribute of length one, whereas the former has no dim attribute.