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First:

By convention, in Python world, the shortcut for numpy is np, so:

In [1]: import numpy as np
In [2]: a = np.array([[1,2],[3,4]])

Second:

In Numpy, dimension, axis/axes, shape are related and sometimes similar concepts:

dimension

In Mathematics/Physics, dimension or dimensionality is informally defined as the minimum number of coordinates needed to specify any point within a space. But in Numpy, according to the numpy doc, it's the same as axis/axes:

In Numpy dimensions are called axes. The number of axes is rank.

In [3]: a.ndim # num of dimensions/axes, *Mathematics definition of dimension*
Out[3]: 2

axis/axes

the nth coordinate to index an array in Numpy. And multidimensional arrays can have one index per axis.

In [4]: a[1,0] # to index `a`, we specific 1 at the first axis and 0 at the second axis.
Out[4]: 3 # which results in 3 (locate at the row 1 and column 0, 0-based index)

shape

describes how many data along each available axis.

In [5]: a.shape
Out[5]: (2, 2) # both the first and second axis have 2 (columns/rows/pages/blocks/...) data