The rank of a matrix is the dimension of the span of its columns. Hence, the rank is less than or equal to the number of columns, i.e. the dimension of the input space.
Intuitively, this kind of makes sense. If I have a $3 \times 2$ matrix like in your example, then as you noted, it can be interpreted as a linear transformation from 2D space to 3D space, say $T: \mathbb{R}^2 \to \mathbb{R}^3$. Since our input space is 2-dimensional, you can think of it as having only "2 vectors" to work with (by which I mean 2 basis vectors, since a 2D space is generated by 2 linearly independent vectors). More precisely, if $v_1, v_2$ is a basis for our 2D space, then the image is going to be $$\text{image of $T$} = \{Tv: v \in \mathbb{R}^2 \} = \{ T(a_1v_1 + a_2v_2): a_1, a_2 \in \mathbb{R} \} = \{a_1T(v_1) + a_2 T(v_2): a_1, a_2 \in \mathbb{R} \} = \text{span of $\{T(v_1), T(v_2)\}$}.$$ Therefore, the image is spanned by two vectors, so certainly its dimension must be less than or equal to 2.
To answer your question: the rank is 2, hence the nullity is 0. In general, the rank and nullity can be found by row-reducing, but in this case, the example is simple enough: there are only two columns, and we can see that neither is in the span of the other, so they are linearly independent, i.e. the rank is 2.
Best Answer
Full row rank means that the rows are linearly independent and full column rank means that the columns are linearly independent.
For a square matrix we say the matrix is full rank if all rows and columns are linearly independent. For a non-square matrix, either the columns or the rows are linearly dependent (whichever is larger).
To say that a non-square matrix is full rank is to usually mean that the row rank and column rank are as high as possible.
In the example in the question there are three columns and two rows. the matrix is full rank if the matrix is full row rank.