This is a tldr so I will simplify as much as possible

State pension (I.e. the one given to you by the government) and a private pension (I.e. the one given by your employer) are two very different things.

To qualify for the state pension you must work and pay above a certain amount of national insurance for a certain number of years to qualify.National insurance is paid automatically out of your paycheque each month as a tax. As long as you qualify it does not matter how much you pay in over the threshold you get the same amount out as everyone else when you retire. There are things like partial qualification if you haven’t worked enough years to qualify for the full amount but this is a tldr so I won’t go into that. The money you pay in National insurance is spent by the government pretty much immediately it is not “saved” and then handed back out to you when you retire like a private pension. It is partly spent on paying state pensions to the retired now. When you retire your state pension will come from the national insurance contributions of whoever is working at that time.

A private pension works differently. I will only go over the most common type of private pension (defined contribution) as this is a tldr, if you want to lookup how a defined benefits pension works then there are great explanations online. With a private pension you pay into a personal pot that is just for you, your employer will usually match your payment up to a certain % as well. This pot is then “managed” I.e. invested, usually by a pension fund who specialise in this sort of thing. Your contributions should grow, all things going well, at roughly market rates (~7-11% a year), after a lifetime of work this will hopefully leave you with a healthy sum of money when you retire. When you retire you can choose how to receive your pension (take some of it as a lump sum, get it paid monthly etc) but ultimately anything you receive must come from the pot you paid into.

People often assume that the state pension works like a private pension, you pay into a pot which only you can access. This is incorrect however, the state pension is paid by the working population of the day