If A is a m by n matrix, the rank of A is the largest number of columns of A that constitute a linearly independent set. This set of columns is not unique, but the cardinality (number of elements) of this set is unique. It's a remarkable fact that rank(A)=rank(A'), i.e. row rank equals to column rank.
Full rank means that rank(A)=min(m,n). For more details, please refer to any linear algebra textbook.