A black hole evaporates by radiating away energy in the form of photons, gravitons, neutrinos, and other elementary particles in a process that involves quantum field theory in curved spacetime. This causes it to lose mass, and so its radius shrinks. It remains a black hole as it shrinks. The increased spacetime curvature at the horizon makes it radiate more and more powerfully; its temperature gets hotter and hotter. The more mass it loses, the faster it loses what it has left!
I agree with Michael Walsby that small black holes are speculative and have not been detected. I am not so sure that they never will be, and it is important to understand how they behave.
As the Wikipedia article explains, for a non-rotating black hole of mass $M$, the radius of the event horizon is
$$R=\frac{2G M}{c^2}$$
and the Hawking temperature is
$$T=\frac{\hbar c^3}{8\pi k_B G M}.$$
If you make the approximation that the black hole is a perfect blackbody, then the radiated power is
$$P=\frac{\hbar c^6}{15360\pi G^2 M^2}$$
and the lifetime of the hole is
$$t=\frac{5120\pi G^2 M^3}{\hbar c^4}.$$
Notice the simple power dependence of all these quantities on $M$. Everything else is just constants. It is easy to substitute numerical values and compute the following table for black holes whose masses range from that of an asteroid down to that of a bowling ball:
$$\begin{array}{ccccc}
M\text{ (kg)} & R\text{ (m)} & T\text{ (K)} & P\text{ (W)} & t \text{ (s)}\\
10^{20} & 1.49\times10^{-7} & 1.23\times10^{3} & 3.56\times10^{-8} & 8.41\times10^{43}\\
10^{19} & 1.49\times10^{-8} & 1.23\times10^{4} & 3.56\times10^{-6} & 8.41\times10^{40}\\
10^{18} & 1.49\times10^{-9} & 1.23\times10^{5} & 3.56\times10^{-4} & 8.41\times10^{37}\\
10^{17} & 1.49\times10^{-10} & 1.23\times10^{6} & 3.56\times10^{-2} & 8.41\times10^{34}\\
10^{16} & 1.49\times10^{-11} & 1.23\times10^{7} & 3.56\times10^{0} & 8.41\times10^{31}\\
10^{15} & 1.49\times10^{-12} & 1.23\times10^{8} & 3.56\times10^{2} & 8.41\times10^{28}\\
10^{14} & 1.49\times10^{-13} & 1.23\times10^{9} & 3.56\times10^{4} & 8.41\times10^{25}\\
10^{13} & 1.49\times10^{-14} & 1.23\times10^{10} & 3.56\times10^{6} & 8.41\times10^{22}\\
10^{12} & 1.49\times10^{-15} & 1.23\times10^{11} & 3.56\times10^{8} & 8.41\times10^{19}\\
10^{11} & 1.49\times10^{-16} & 1.23\times10^{12} & 3.56\times10^{10} & 8.41\times10^{16}\\
10^{10} & 1.49\times10^{-17} & 1.23\times10^{13} & 3.56\times10^{12} & 8.41\times10^{13}\\
10^{9} & 1.49\times10^{-18} & 1.23\times10^{14} & 3.56\times10^{14} & 8.41\times10^{10}\\
10^{8} & 1.49\times10^{-19} & 1.23\times10^{15} & 3.56\times10^{16} & 8.41\times10^{7}\\
10^{7} & 1.49\times10^{-20} & 1.23\times10^{16} & 3.56\times10^{18} & 8.41\times10^{4}\\
10^{6} & 1.49\times10^{-21} & 1.23\times10^{17} & 3.56\times10^{20} & 8.41\times10^{1}\\
10^{5} & 1.49\times10^{-22} & 1.23\times10^{18} & 3.56\times10^{22} & 8.41\times10^{-2}\\
10^{4} & 1.49\times10^{-23} & 1.23\times10^{19} & 3.56\times10^{24} & 8.41\times10^{-5}\\
10^{3} & 1.49\times10^{-24} & 1.23\times10^{20} & 3.56\times10^{26} & 8.41\times10^{-8}\\
10^{2} & 1.49\times10^{-25} & 1.23\times10^{21} & 3.56\times10^{28} & 8.41\times10^{-11}\\
10^{1} & 1.49\times10^{-26} & 1.23\times10^{22} & 3.56\times10^{30} & 8.41\times10^{-14}\\
10^{0} & 1.49\times10^{-27} & 1.23\times10^{23} & 3.56\times10^{32} & 8.41\times10^{-17}\\
\end{array}$$
As you can see, as the hole shrinks, it gets tremendously hot and radiates enormous amounts of power. This is why Hawking titled one of his papers "Black hole explosions?"
As far as I know, no one is sure whether a hole evaporates completely or leaves behind a Planck-scale remnant.