In general, the DFT is arduous to compute. It requires $N^2$ multiplications ($N$ multiplications between each sample in $x[n]$ and $\textrm{exp}(-j n {(2 \pi)}/{(N)} k)$ for each of the $N$ terms in the sum). Therefore, as $N$ grows, the number of multiplications grows very fast.
The fast Fourier transform (FFT) allows us to compute the DFT much quicker. The original Fast Fourier Transform algorithm (the radix-2 Cooley-Tukey algorithm) requires only $(N/2) \log_2(N)$ multiplications. Therefore, if $N = 1024$, computing the DFT requires $1,048,576$ multiplications while the radix-2 Cooley-Tukey algorithm requires only $5,120$ multiplications. That is a $204.8$ times reduction in computation.
The FFT works by realizing that we can represent the DFT of size $N$ as $$ \begin{eqnarray*} X[k] &=& \displaystyle DFT_{N} \left\{ x[n] \right\} \\ &=& \sum_{n = 0}^{N-1} x[n] e^{-j n \frac{2 \pi}{N} k} \\ &=& \sum_{n = 0}^{N/2-1} x[2 n] e^{-j (2 n) \frac{2 \pi}{N} k} + e^{-j \frac{2 \pi}{N} k} \sum_{n = 0}^{N/2-1} x[2 n + 1] e^{-j (2n) \frac{2 \pi}{N} k} \end{eqnarray*} $$ This splits the DFT into two sets of terms: even indices plus the odd indices. If we do a few more manipulations, $$ \begin{eqnarray*} X[k] &=& \sum_{n = 0}^{N/2-1} x[2 n] e^{- j (2 n) \frac{2 \pi}{N} k} + e^{-j \frac{2 \pi}{N} k} \sum_{n = 0}^{N/2-1} x[2 n + 1] e^{-j (2n+1) \frac{2 \pi}{N} k} \\ &=& \sum_{n = 0}^{N/2-1} x[2 n] e^{- j n \frac{2 \pi}{N/2} k} + e^{-j \frac{2 \pi}{N} k} \sum_{n = 0}^{N/2-1} x[2 n + 1] e^{-j n \frac{2 \pi}{N/2} k} \\ &=& DFT_{N/2}\left\{ x[2n] \right\} + e^{- j \frac{2 \pi}{N} k} DFT_{N/2}\left\{ x[2n + 1] \right\} \\ &=& H[k] + e^{- j \frac{2 \pi}{N} k} G[k] \end{eqnarray*} $$ Therefore, we can represent the DFT of length $N$ as the sum of two DFTs of length $N/2$ ($H[k]$ and $G[k]$).
Since $H[k]$ and $G[k]$ are the result of DFTs of size $N/2$ they have periodicity of $N/2$. Yet, the complex exponential in front of $G[k]$ has a period of $N$. Since the complex exponential factor has the relationship (since $k$ is an integer) $$e^{j \frac{2 \pi}{N} (k + N/2)} = e^{j \frac{2 \pi}{N} k} e^{j \pi k} = -e^{j \frac{2 \pi}{N} k} \; ,$$ we can represent the DFT as $$ \begin{eqnarray*} X[k] &=& H[k] + e^{- j \frac{2 \pi}{N} k} G[k] \quad , \quad 0 \leq k \leq N/2 - 1 \\ X[k+N/2] &=& H[k] - e^{- j \frac{2 \pi}{N} k} G[k] \quad , \quad 0 \leq k \leq N/2 - 1 \end{eqnarray*} $$ This gives us the full $N$ DFT result. Therefore, we can compute two DFTs (of length $N/2$) and then combine the results to get complete DFT (of length $N$). If the relationship between every value is drawn in a diagram, we get a "butterfly" structure of additions and multiplications.
We can now further break the two $N/2$ size DFTs in four $N/4$ size DFTs and then further into eight $N/8$ size DFTs. We can repeat this step until we can no longer divide the DFT size by 2. This fully decomposed representation is the radix-2 Cooley-Tukey algorithm.