To claim unemployment is to take advantage of Unemployment Insurance. Companies pay into that and when you are terminated (doesn't count if fired for cause) you get some predefined amount of money for a period of time to help while job searching. The amount is dramatically less than your normal pay.
Unemployment insurance is a form of insurance in US that pays you if you lose your job due to layoffs and other conditions. It is paid by you so it is not “welfare.” It is managed by each state independently and has some rules. For example, if you lose you job today and “don’t claim” unemployment insurance for two weeks, those two weeks are not paid out to you (this is in Texas). Every week, you must show that you are “actively” looking for a job. You may lose the benefit if you are not, or if a job is available but you decide not to take it. You must report on your efforts to find a job (interviews etc), and must attend some state class about looking for work.
Most states’ unemployment insurance covers up to a maximum benefit per week, and a maximum number of weeks. When I lost my job once, I was paid $325 per week.
At least in my state, the employee does not pay unemployment insurance tax, but the amount the employer pays is usually reported on the paystub. That is, if you are paid $85,000 salary it won't be reduced by the unemployment tax. However, some economists treat all payroll taxes as effectively paid by the employee out of "total compensation".
