You can definitely install custom software :-).
A Droplet is a VPS (Virtual Private Server), which provides you with full
root access, thus you're able to install any software you'd like (as long as it can be installed on a Linux OS).
You're essentially provided with a blank canvas from which to build to suite. You'll start out with a bare OS and from there, you can install the software you need using the package manager or other repos.
One key difference between a Droplet (VPS) and shared hosting is that you need to be comfortable working from the command line (CLI), at least from the start. There's no control panel (such as cPanel) to help set these things up. Of course, there is the community, so if you have any questions along the way, feel free to ask!