What firmwares can I install Homebrew on?
For the end user, you can't really do much yet. There's PegaSwitch, but that's really meant for devs.
For the future, however, there has been a major exploit announced for firmware versions 1.0.0 to 3.0.0, dubbed smhax. This exploit gives full permissions to any service.
What is smhax?
smhax is most likely one of the biggest vulnerabilities found in the Switch firmware, and will probably be the main access to homebrew for a while.
Nintendo have since patched smhax in Switch firmware 3.0.1, so it's heavily advised that you DO NOT UPDATE!
smhax, when fully exploited, will allow privilege escalation which will allow us to register and run arbitrary services on the console.
Prior to 3.0.1, the service manager (sm) built-in system module treats a user as though it has full permissions if the user creates a new “sm:” port session but bypasses initialization. This is due to the other sm commands skipping the service ACL check for Pids <= 7 (i.e. all kernel bundled modules) and that skipping the initialization command leaves the Pid field uninitialized. Successful exploitation results in Acquisition, registering, and unregistering of arbitrary services
When a homebrew entrypoint is found, smhax would likely be used to gain access to the whole console.
I'm on 3.0.1 or above! Can I downgrade?
Unfortunately, with the Switch, it has become extremely hard to downgrade. Nintendo have learned from their mistakes and introduced a new security measure: eFuses.
What's an eFuse?
An eFuse is a tiny element inside the CPU which can be "burnt" to permanently change the state of the CPU. For example, whenever you upgrade your Switch firmware, one of these eFuses is burnt. When you updated to 3.0.1, another of these eFuses burnt.
Because of this, if you tried to downgrade the console to 3.0.0, the console would notice that an extra eFuse is burnt than normal, meaning it knows it's previously been on higher firmware, and if too many eFuses are burnt, the bootloader will panic immediately and cease all other actions.
Can't we just do a downgrade with a hardmod?
Sadly not. The bootloader would still notice that too many eFuses have been burnt, and begin to panic.
Does this mean we'll never be able to downgrade on the Switch?
Not necessarily. While it may be impossible to downgrade at this point in time, in the future with more advanced exploits, this may be possible. However, for now, you'll just have to wait on 3.0.1 or above for another exploit.
I want to play the newer games. Should I update?
Well, I can't stop you, however SciresM stated that "if you care about hax, do not update" and that staying on 3.0.0 or below is "the correct call". He also said that people updating for newer games "will likely be waiting a much, much longer time".