A register is a part of the processor that holds a bit pattern. Processors have many registers.
MIPS processors have 32 general purpose registers, each holding 32 bits. They also have registers other than general purpose ones.
The first example SPIM program puts bit patterns representing integers into two registers. Then it adds the two patterns together. The newest version of Spim is called "QtSpim". If you have not installed it already, do so by following the instructions in the appendix.
The picture on the right shows how to start SPIM on Windows 8. You can press the Start menu and search for "QtSpim". The first result should be the QtSpim application.