Most workers employed on a full-time, permanent basis have unemployment insurance paid on their behalf by employers. Economists generally agree that the costs of the insurance are ultimately borne by workers. If someone is involuntarily laid off, they receive up to (I think) 26 weeks of UI. The share of unemployed workers who are eligible for UI benefits is around 1/3 (if memory serves me right, but it's definitely a fraction).
Each state organizes a UI trust fund. During recessions, the Federal government usually boosts that fund and allows states to extend their UI benefits; at present, workers in states with the highest unemployment levels can collect for up to 99 weeks. That may seem like a long time, but when you have 5 or more workers for each job opening, it can take people a year or more to get re-employed. UI is probably one of the biggest short-term boosters for the economy, because people receiving UI benefits generally spend them all on necessities. That extra money replaces demand for goods and services that would otherwise have been lost.
Lately, there has been some hysteria among conservatives that UI reduces the incentive for people to find work. Opponents have wondered if it's becoming a permanent entitlement. Senator James Bunning, a Republican from Kentucky, recently held up an extension to UI coverage in the U.S. Senate. He also cited disbelief that UI boosts the economy.
Unemployment insurance is a critical social insurance provision that provides some support to people who are between jobs and enables them to keep looking. In recessions, it is also a core countercyclical economic booster. We can debate later if eligibility levels should be scaled back - and they certainly don't need to be as long as 99 weeks in normal times - but it is insane to oppose extending unemployment insurance when people are unable to find jobs to replace the ones they lost.