There are a number of audio plugin standards, and they vary across operating systems and in what DAWs support them. Here's an overview of the various types of plugin.
VST, short for Virtual Studio Technology, was the first plugin standard. It was invented by Steinberg and rose to prominence with the Cubase digital audio workstation. Today it is the most popular, with the vast majority of plugins being VSTs.
There are two major versions of the VST standard. The first is VST2, and it has been discontinued and applications are gradually removing support for it. However, a vast amount of software has been written as VST2, and some of it cannot be updated, since the original developer is not around.
The VST3 standard is the current version, and has the widest support. VST2 and VST3 plugins are not cross-compatible, and supporting one does not automatically give you the other.
There are also extensions to VST called ARA (Audio Random Access) that allow a DAW to communicate more information to a plugin. This was developed by Celemony and PreSonus in order to enhance the capabilities of Melodyne.
AU plugins are similar to VSTs, but are only supported on Apple devices and are part of the Core Audio system. The ARA extensions also work with AU plugins.
LADSPA is a Linux-based audio plugin standard that is a bit more simplified than other standards. Although there is still much software that supports LADSPA and many plugins written with it, the LV2 standard (LADSPA version 2) has many more capabilities and should be considered the modern standard on Linux.
DSSI, the Disposable Soft Synth Interface, is a Linux-based standard for writing virtual instruments. The LV2 standard now includes support for virtual instruments, and unifies DSSI and LADSPA into a single plugin standard.
CLAP stands for CLever Audio Plug-in and is a new open-source standard developed by Bitwig and u-he. It is designed to overcome both the limitations of plugin standards such as AU and VST, and also the proprietary licensing issues.
RTAS stands for Real-Time AudioSuite and is an older proprietary plugin format that was used by Pro Tools. It has been superseded by AAX and Pro Tools support for RTAS was removed in 2013.
AAX stands for Avid Audio eXtension and is the modern proprietary plugin format used by Pro Tools.