A jockstrap is a kind of undergarment for men intended for athletic use. The garment consists of a waistband and a pouch in front in which the penis and scrotum/testicles are worn, connected under the legs to the waistband in back. Jockstraps typically do not have a seat. The pouch is typically from a highly stretchy, resilient, supportive material since the point to a jockstrap is to give extra support to the genitals during physical activity and thereby reduce or prevent injury.
Hockey Jock Shorts are usually a loose mesh pair of shorts that have a jock with cup built into, or sewn into them. They also have sections of velcro on the front and back of the lower leg section that are used for affixing your hockey socks to. Without jock shorts you have to use hockey tape or a garter to hold up your socks. Compression shorts are like hockey jock shorts except they fit tightly and are usually made out of a different type of material. An example of the hockey jock short is the ITECH JM200 Competitive Jock Short. An example of compression shorts is the ITECH JC200 Compression Jock Short.
Jock straps come in two different types. Some have a pouch to hold a hard cup in place, for sports where you might get hit. A soft jock strap is just designed to hold the genitals up very tightly against the body so they don't bounce and twist during athletics.
Some of them have a pouch, and can hold a hard plastic cup to keep you from getting hit in the genitals. Others are made to keep your penis and testicles tightly in place so they don't bounce when you run or work out.
A jockstrap is what holds the cup (plastic shell to protect your twig and berries) in place over your crotch.