Grasses and legumes are two different types of forage made into hay for horses. Grass hays are generally easier to grow than alfalfa but also lower in protein and higher in carbohydrates. The most common types of grass fed to horses in the US are timothy, brome, bermuda, and orchard grass.
Alfalfa hay is made from the legume alfalfa and like all legume hays, it is high in protein and is usually very palatable. Due to its high protein content, it has a high feed value and thus is especially useful for young horses, pregnant mares, and lactating mares. The protein is not always useable and over feeding alfalfa will result in a high urine output. It needs to be fed with discretion and ideally should be fed along with grass hay.
One of its benefits is that being a legume it does not have the seed cycle that grasses have, so it can be cut several times a year. Each cut will have roughly the same feed value. The leaves of the alfalfa are tiny, but this is where most of the nutrition is, so if there is extensive leaf drop it can lose feed value.
The stalks can be hard and tough if the plant is too old at the time of hay making. Due to weather constraints, the first cut of the year may be a stalkier, less nutritious cut. It is a very useful feed for horses but needs to be fed with discretion.