They've gotten so embedded in pop culture that they've almost become cliché, but depending on the social media platform you're using, hashtags can be invaluable. If you're not familiar with what a hashtag is, it's the same symbol seen on a standard touch dial phone button that is the "pound" or "number", and denoted by #.
At the moment, hashtags are by far most relevant on Instagram and Twitter. Facebook as a platform does make use of them, but most of Facebook users do not. Chalk that up to chaos theory, or that they enabled them a lot later in their life-cycle than the other two, as to why that is, but they are no where near as commonly seen on Facebook, currently.
While they were popularized by Twitter, currently, Instagram seems to make the most use of hashtags (or simply "tags", as they refer to them), and is currently the primary way of searching for photos on it.
The format for hashtags are pretty straight-forward. For example, if you want to post about something related to social media, you'd include it written either as: #socialmedia, or #SocialMedia, and people specifically looking for posts related to social media could potentially find your post. While capitalization doesn't matter, a space between the words will not work. So, if you do want to differentiate between a number of words, you can use capital letters.
Historically on Twitter, the most popular and relevant hashtags were all that used to be featured in their "Trends" section, though, currently, it's about evenly split between keywords (actual, unformatted words) and hashtags. If you use a hashtag in your tweet that's currently trending, it's more likely to be seen by a higher number of people, and in turn, interacted with by more people, so long as the tweet is relevant to the hashtag.
The platform where hashtags can make the most difference right now is on Instagram, where it could be the difference between your photo getting a significant amount of interaction, or none. If you don't use a hashtag, the only people likely to see your photo will be those on your friends list, and perhaps, their friends, if and when someone interacts with it. On the other hand, if you make appropriate use of hashtags (making sure to include only tags that are relevant to the photo, itself), anyone on Instagram searching for that tag at the moment could potentially see it - which significantly increases the size of your potential audience. What's more, it will primarily appear to people who are specifically interested in it.
We've seen people with amazing photos that have absolutely no interaction, just because they failed to use hashtags. Some people are "too cool" to use them, but if you want your photos seen by the most amount of people, we highly recommend using them, and as many as are relevant, at least while you're initially establishing your following. The current limit for number of hashtags you can include on an Instagram post is 30, so, make them count!