You can use the .shape property or just len(DataFrame.index). However, there are notable performance differences ( len(DataFrame.index) is fastest):

In [1]: import numpy as np

In [2]: import pandas as pd

In [3]: df = pd.DataFrame(np.arange(12).reshape(4,3))

In [4]: df
Out[4]: 
   0  1  2
0  0  1  2
1  3  4  5
2  6  7  8
3  9  10 11

In [5]: df.shape
Out[5]: (4, 3)

In [6]: timeit df.shape
2.77 µs ± 644 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

In [7]: timeit df[0].count()
348 µs ± 1.31 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

In [8]: len(df.index)
Out[8]: 4

In [9]: timeit len(df.index)
990 ns ± 4.97 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

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EDIT: As @Dan Allen noted in the comments len(df.index) and df[0].count() are not interchangeable as count excludes NaNs,