You could save money by eating only cat food or performing your own dental surgery, but you probably should not try either of those. It's also possible to save money by giving up things you love, and, generally, that's not a best practice either.
Instead, saving money involves making sensible cuts, understanding your spending, and becoming more mindful of your money. You don't have to sacrifice everything you love or start cutting your own hair. There's a reasonable middle for everyone.
1. Have a budget
It's very hard to save money if you don't know where your money is going. Your budget does not need to be overly complicated, but you need to have one. Start with big expenses (rent/mortgage, car loan, student loan, etc.) and work your way down to the smaller ones. It's OK to group some items into a category like "eating out," but the more detail you have, the easier it will be to make decisions as to what can be cut.
2. Trim the fat
If you look at your budget, you may realize you're spending money on things that don't matter to you. Laziness may cause you to buy a $5 coffee every day when you'd be just as happy with the shared pot in the office. Maybe you're paying for cable channels you don't watch or subscription services you barely use. If you don't care much, then these are places to cut back on spending.
3. Own things longer
If you have a $300 monthly car payment and keep your car for an extra year after the loan has ended, that's $3,600 in your pocket. The savings won't be as big for smaller items, but if you can get another season out of your winter boots or put off buying a new couch, you can have a little more cash to put away.
4. Do some things a little less
Saving money sometimes involves sacrifices. That does not mean you have to give things up entirely. If you eat out a lot, for example, you might consider cutting back by 25%. The same could be said for your travel and entertainment budgets. Don't give up what you love, just do it a little bit less.
5. Put it on automatic
It's hard to spend money that's not in your checking account. Set up an automatic weekly or monthly transfer from your main account to a brokerage account (or even a savings account). Even small amounts of money can pile up quickly if you invest steadily and make saving something you do on autopilot.
It's a lot of little things
Everything above assumes you're in decent financial health in the first place. If you're not or have big plans for spending (like saving up for a down payment on a house) you may need to be a little (or a lot) harsher to meet your goals. For people in good financial shape, however, it's easy enough to save more by making lots of little changes -- some of which you may not even notice.