I've been using Ubuntu for a while and I want to switch to elementary.
The problem is that I don't know if I can install Ubuntu applications on elementary.
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TL;DR: It depends, but mostly yes.
For a the vast majority of software, you'll be fine. Apps like LibreOffice, Firefox, Gimp, Transmission, etc all run on elementary OS. Pretty much anything you can get from Software Center will run.
However, there are some apps that have been built specifically for Ubuntu and will not run. The Intel Graphics Driver Installer will refuse to run on anything that isn't explicitly Ubuntu. Installing Unity Tweak Tool can actually have negative effects on your system (like changing your lock screen) and almost none of the features it provides work with elementary OS.
Some apps conflict with elementary OS's default apps. For example Shotwell can't be installed side-by-side with Photos.
Some apps expect desktop environment features that aren't present in elementary OS. For example, Skype and Dropbox expect a specific API to display their panel icons which isn't present.
If the developer doesn't say they support elementary OS on their website, ask them. Many developers are happy to make sure their apps behave as expected, but they may be unaware of the demand for them to run on environments they haven't used themselves.