The difference between bracket [ ] and double bracket [[ ]] for accessing the elements of a list or dataframe

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The R Language Definition is handy for answering these types of questions:

R has three basic indexing operators, with syntax displayed by the following examples 

    x[i]
    x[i, j]
    x[[i]]
    x[[i, j]]
    x$a
    x$"a"

For vectors and matrices the [[ forms are rarely used, although they have some slight semantic differences from the [ form (e.g. it drops any names or dimnames attribute, and that partial matching is used for character indices). When indexing multi-dimensional structures with a single index, x[[i]] or x[i] will return the ith sequential element of x.

For lists, one generally uses [[ to select any single element, whereas [ returns a list of the selected elements.

The [[ form allows only a single element to be selected using integer or character indices, whereas [ allows indexing by vectors. Note though that for a list, the index can be a vector and each element of the vector is applied in turn to the list, the selected component, the selected component of that component, and so on. The result is still a single element.

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