Wages seem much lower in Spain but on the other hand it seems like you've got a much more sturdy safety net in retirement.
Retirement in the US is really complicated. Your retirement funds are a mix of private and public funds. Every paycheck gets a small portion taken out for Medicare and for Social Security. Medicare is the public health insurance system for everyone 65 years or older. It's like the coverage you guys in Europe get. Social Security is the public pension fund. The amount you receive in retirement depends both on how much you put in (which is based on what you earn) and the age at which you choose to start receiving the benefits. You can start receiving Social Security at 62, but you receive a smaller amount of money. If you wait till you are 65, 68, or 70 to start receiving, you get a much larger amount per month. So if you continue working until a later age (or have enough money from private savings or investments to hold you over) you can make a lot more money when you do start taking the public funds. Also, once you retire and start taking Social Security, you're limited income per year you can earn from a job. You'll see a lot of people that work part time to supplement their social security income.
The other part of retirement is from private funds. Some companies have their own private pension plans, but most places these days have what's called a 401k plan where a % of your gross income gets taken out of your pay and put into a diversified investment fund. Your company then contributes a matching percentage for free - for example, 4% of my pay is invested in my company's fund, and my company adds in an extra 4% on top. If I ever change companies, I can take the money invested and roll it over into a new company's plan, or into a personal plan.
You can also invest on your own in mutual funds and stocks which have no limit, and in special funds called IRAs - Individual Retirement Accounts. With these accounts you can invest a set amount per year (this year it is $6500, but it usually goes up every year or so) and there are tax incentives for doing so.
So all in all it's quite a bit more complicated than just having a government-run public pension system but there's definitely the opportunity to make some money and set yourself up for a nice retirement. If I continue making the maximum IRA contribution every year until I retire at 65 (and never make a withdrawal), with an average rate of return of 5%, that account will have roughly 850k with me having contributed 260k.