Spectral indices are a set of tools that allow us to highlight certain features from multispectral and hyperspectral reflectance data.
At their most simple, they are mathematic equations that use individual wavebands or wavelength reflectance to create a new layer that will provide us with new information about the selected wavelengths.
There are a huge amount of spectral indices - one that you are likely to be familiar with is the Normalised Difference Vegetation Index, or NDVI.
This uses the Near Infrared and Red wavebands to create a measure of green vegetation. This helps us to differentiate between green vegetation, and things that just appear green when looking at a RGB representation of a multispectral or hyperspectral image. The green might be a leaf, or grass - but it might also be a bit of green paint, plastic, or anything else. Spectral indices help us to pick out the extra information that multi and hyperspectral datasets provide.
NDVI is just one tool in a huge box - it is just one of a large number of indices that look at 'broadband greenness'. Each have their advantages and disadvantages, and you must take care that the SI that you employ is suitable to both your dataset, and the output you require.