There are several types of leave in the US military. The most common is the 30 days a year that everyone earns. It is basically like vacation time in the civilian world, but instead of being in hours, it is in days and is also charged for days that the person normally wouldn't be working (like weekends and holidays).
Convalescent leave is what would be a 'sick day' in the civilian world. The big difference is that it is unlimited. It usually requires that someone be seen by a doctor who recommends time off.
Passes are something that commanders can give for whatever reason they want. A person can't be on a pass for more than 96 hours--any longer and they have to use leave time. Technically any time people get 4 day weekends for a holiday, they are on a pass.
Servicemembers are on duty 24/7. A 'workday' could be anything from a couple of hours to the entire day. They don't clock in and out. So, if someone needs to run an errand, get a haircut, meet with their kid's teacher, they can, as long as their boss is OK with it.