In the following pair, which one is more basic and why?
$C{{H}_{3}}N{{H}_{2}}$ or ${{C}_{6}}{{H}_{5}}N{{H}_{2}}$

Hint: Think about what having a basic character means with respect to electrons the electrons. Visualize the structures of both the given molecules and how the electrons act within the molecule.

Complete step by step answer:
Basicity is defined as the ability of any atom, group, or molecule to donate electrons. This is usually assumed if a molecule has atoms that has some lone pairs of electrons that can be donated. In the amino group, the lone pair that is present on the nitrogen atom is considered to contribute to the basic character of the group. Now, let us look at the structures of both the given molecules to determine which nitrogen can use the lone pair for bonding easily.

Here, methyl amine has the molecular formula $C{{H}_{3}}N{{H}_{2}}$ and aniline has the molecular formula ${{C}_{6}}{{H}_{5}}N{{H}_{2}}$.

Let us consider aniline. We know that benzene has several resonating structures for stability. The lone pair that is present on the nitrogen atom in the aniline molecule delocalize and contribute to the stabilization of the aromatic ring. The nitrogen atom donates its lone pair to form a bond with the aromatic ring and gains a temporary positive charge. When this happens, it does not have any more electrons that it can donate to form any bonds which contribute to its basic character. Essentially, due to delocalization of electrons from the nitrogen atom, it loses its basic character. The resonating structures of aniline are:

Here, we can see that the lone pair of electrons on the nitrogen atom is not available for most of the resonating structures. Thus, this reduces the basic character of aniline.
Now, let us consider the structure of methyl amine. There is no aromatic ring that requires stabilizing and thus no delocalization of the lone pair on the nitrogen atom occurs. Thus, the basic character of the amino group is retained.
Hence, methyl amine ($C{{H}_{3}}N{{H}_{2}}$) is more basic than aniline (${{C}_{6}}{{H}_{5}}N{{H}_{2}}$)

Note: Remember that the methyl group is an electron donating group. So, the electron density around the nitrogen atom increases and makes it even easier to donate the electrons.

Here, we can see the partial positive and negative charges present due to the inductive effect.