If you ever come across a kitty without a home, it is important you know what you should (and shouldn't) do to make sure they are safe. The first important factor is to know the difference between a stray cat and a feral cat. A feral cat has always lived apart from humans and is considered 'wild', whilst a stray cat has previously had a home with a human but due to unfortunate circumstances no longer does. There are many reasons that a cat can become a stray and it is important not to assume that they are unwanted.
If the cat does not appear to be in any distress or danger, you should first use a paper collar. A paper collar allows you to write a message asking the cat's owners to contact you so that you can establish whether the cat is in fact a stray. Cats Protection have a great article showing you exactly how to make a paper collar and how to use one safely here. They also have a printable template here. It's also a good idea to post on local Facebook groups, missing pet websites, across social media or even leave posters with local businesses to spread the word that you have found a cat.
If you get no response after this, you should report the stray cat to your local rescue shelter and/or vets. Many vets and shelters keep a lost and found database so they may already have an owner's details if they are looking for their cat. If you are able to approach and catch the cat then it is a great idea to take them to your local vet so that they can scan them for a microchip. After this, it may be time to take the cat to a local rescue centre.
Unlike dogs, local authorities do not take in stray cats, so you will have to find a shelter that has the space to take in the cat. Once you have found a shelter that can take them, drop the cat off and the shelter will take care of the rest. If you decide to keep the cat, make sure you fully consider the commitment and responsibility of owning a pet so that the cat doesn't end up without a home again. If you find the cat's owner please remember to inform anyone who may still be looking and take down any found posters you put up.