American here. Checks are basically little pieces of paper (about the size of two credit cards; they usually come in a book of 25-50) that allow the recipient to make a demand against my account. I write on the check the name of the recipient, the date, the amount (in two places, one in long form and one in numeric form) and sign the bottom. They have a set amount of time in which they can deposit the check (I think it's somewhere between 30 and 90 days, but I can't remember). The recipient endorses it on the back by signing it and deposits it at their bank, and their bank handles transferring the money between the banks and into the appropriate accounts.
Printed on the bottom of the check is my account number and a routing number for my bank. They're printed in a format called MICR, which uses magnetic ink (or used to, not sure if they still do) so that computers can scan them quickly. When I worked at a grocery store in the 90s, our systems were supposed to actually verify whether or not a check was "good" by checking the amount against the amount available in their accounts. Not sure if this really worked, because I never saw a check get rejected.
It used to take several days for the whole thing to play out. What would happen is that, when the recipient deposited one of my checks, their bank would credit them the amount of the check immediately (there are some limitations on this depending on the size of the check; banks have various policies about that) and the check would enter the inter-bank system. A day or so later, the actual amount would be withdrawn from my account.
In most places, passing a bad check (called "bouncing") is a crime, though most banks will provide some type of overdraft protection (usually with an accompanying steep fee) to keep you from getting in trouble with the law.
Checks were widely used up until the early-mid 2000s, when debit cards started to become widespread. These days, they are primarily used by the elderly who refuse to adopt newer technology, in businesses that do not use direct deposit, and in complex financial situations (such as purchasing real estate). Other than that, the vast majority of purchases in America happen via the card networks.
I write maybe three checks a year and receive maybe two. Everything else is done online or by the card networks. And best of all, when I receive a check, my bank (USAA) lets me just take a picture of it with my iPhone and immediately credits me the amount, as a scan is considered a valid reproduction of the check.