Older generation Americans used to work for companies such as GM that had defined benefit plans or DB plans. The companies promised to pay their retirement benefits in the form of pensions. The problem now is that some of these companies are broke and many of their plans are underfunded and the people who have contributed their hard earned money, now face less pension than they were promised.
Government control over this is via PBGC Pension Benefits Guarantee Corporation. It guarantees the pension benefits up to a certain limit.
In the present times, most firms offer 401K plan which is a defined contribution plan or DC plan. The difference here is that you the worker have more control over your future retirement funds. Along with control comes responsibility. i.e. The worker has to make the contributions and invest prudently. This takes some effort and education on the part of the worker.
When can you retire? If you have enough accumulation of wealth in your retirement plan, that it can pay your bills for the rest of your life, you can retire. This is more likely to happen if you start early, contribute consistently and invest prudently.
Teachers are state employees. They have a 403b plan, which is similar to a 401K plan. It is a defined contribution plan.