A watermark is a superimposed graphic, logo, or text placed on an image as a way to identify the photographer/image creator.
Believe it or not, whether to add a watermark or logo to your online photographs is quite a touchy subject to many people. Right up front I am happy to declare myself firmly in the "no" camp - except in specific circumstances. No matter how subtle or discrete, watermarks always detract from the photograph - no exceptions. If you have a beautiful photo to post do you think there is anyway in which that photo could be improved by adding an advertising message. Your watermark is essentially an ad.
There are several reasons why a person may want to add to add a watermark but most of them are not reason enough to actually do it. Let's look at some of those reasons and consider the pros and cons of each:
Watermarks protect images from theft
Possibly, but only from the laziest of thieves. Removing watermarks, either by cropping the image or by cloning it out is pretty trivial these days. Incompetent or unaware photo thieves will probably just use your photo with the logo intact.
In order to really prevent photo theft your watermark needs to be so intrusive as to make it not worth the effort to remove. If not, any self-respecting photo thief will probably just remove your watermark and then add their own.
It makes your image copyright protected
Your images are copyright as soon as you press the shutter button. Adding a logo to the image has no effect on ownership. A better idea is to ensure that your contact details are included in the image's Metadata - most imaging software will have a function to do this.
Logos are great for branding/recognition
This isn't a bad argument, if other people are posting or sharing your photos. If you're posting on your own website/blog/Instagram/Facebook page etc. there is very little benefit because the viewer is already on your page.
Watermarks look professional
Actually, I think the opposite is more correct. Can you think of any accomplished/high-end photographers that use watermarks? You just don't see watermarks on quality art, portrait, fashion, documentary, commercial photographs. Pros don't watermark, newbies do. Your photos are less likely to be featured or published if they contain advertising logos.
Branding digital proofs
This is the one use case that I can think of in which watermarking is a good idea.
Let's say you have been contracted to photograph an event or a portrait session. Your agreement with the client is to provide a certain number of edited, high resolution images and a bunch of unedited, low resolution photos. It's a very good idea to watermark the low resolution shots. It means that your client won't be able to print the lo-res shots but can still use them on their social media pages. Personally, I would not want anyone making prints from my lo-res proofs but I'm happy fro them to buy extra high resolution files.
If you are selling images online you may want to include a logo on the website proof. It should go without saying, you would remove the logo from the purchased file. Nobody is going to pay for a photo with an ad in the corner.
Watermarks are a distraction
To repeat a statement from the top of this post... No matter how subtle or discrete, watermarks always detract from the photograph - no exceptions.
You simply cannot improve a photo by including an advertisement.
Once you have noticed someone's watermark on a photo you might find that your eye is drawn to the logo more than it is to the subject; particularly if the logo is quite bright in relation to the rest of the image.