A linear code with parameters $[n, k, n-k+1]$ is called a maximum distance separable (MDS for short) code. A linear code with parameters $[n, k, n-k]$ is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a $[q+1, 2u-1, q-2u+3]$ MDS code for each $u$ with $1 \leq u \leq \lfloor\frac{q+1}2\rfloor$, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.