You can emulate anything!
Well, nearly anything. As long as it's not modern/complicated enough and was pretty popular. Keep in mind that emulators are written by people for free in their free time - they have their share of issues, quirks and not-so-obvious problems which you may eventually hit. Most older platforms are now emulated well enough to not bother you, but PS2, GameCube and Wii are still quite problematic.
Take a look at this list of OS emulators. Now take a look at the list of game system emulators. Any application or game for any of these systems can be emulated on your PC. You just need to set the emulator up and find the games and applications. The websites and guides below should be helpful, and so will Google.
Most emulators can be used perfectly fine just with mouse and keyboard (peek into their options and user guides to figure out the default keybinds), yet some need a controller with analog sticks (PS2/3, GameCube) or a console-specific input (Wii).
Which emulator should I use for <popular platform>?
Old PC games: Some really old games have major compatibility problems with modern PCs (or rather, their operating systems), especially if they're 16-bit. For these games, DOSBox allows you to easily emulate DOS and play them. Note that not all games will emulate perfectly well - there's a compatibility list (if your game isn't listed there, don't worry - try it and it might work just fine). Now, DOSBox, just like DOS, doesn't have a nice GUI, so you'll have to deal with the command line if you're not getting the game from GOG, where they zeroconf everything for you. Here's a nice tutorial to get you started.
Sony PlayStation: The most popular one is ePSXe. The problem is, you need PSX BIOS. Which we can't help you with - you can't legally download it, but you can dump one from your own PSX. Once you've done that, there's a fairly up to date guide you need to follow before being able to run any games.
Sony PlayStation 2: PCSX2 is the way to go. A quick Getting Started is available on the official page, which you should follow. Again, you'll need a BIOS file, which you can't legally download, but dumping it from your console is fine. Requires a moderately powerful PC.
Sega Genesis: Fusion and Gens seem to be the popular choice.
Nintendo Wii U: Yes, that's right. A perfectly functional Wii U emulator already exists three years after the release of the console it's emulating. Much like Dolphin, CEMU grew very quickly. It's still young, but there are a handful of commercial titles it can play well enough, and it's getting better by the month, especially with the release of the Vulkan API, which is expected to also boost PS4 and Xbox One emulation progress. Go to Cemu.info.
Nintendo GameCube/Wii: There's Dolphin. As GameCube and Wii are similiar, this emulator supports both consoles (and something called Triforce, which you've likely never heard of - don't worry, neither did I). Go for the automated builds, development is very active and you'll be getting massive improvements every day. Requires a moderately powerful PC (some games are extremely demanding, and might not be playable at all without a massively powerful CPU/GPU). Doesn't require BIOS files, however it's able to run Wii System Menu (you may get some free official software with NUS Downloader), and if you're looking to play Wii games, you'll probably need a Wiimote (which you can easily connect to your PC). All the guides are here.
Nintendo Entertainment System/Famicom: FCEUX. Fast and requires no additional setup.
Super Nintendo Entertainment System: SNES9x should be good. Since it doesn't try to be a completely ideal emulator, and often values speed over accuracy (unlike Higan, which has very high requirements), most games work fine and fast even on really old hardware. Requires no special setup.
Nintendo GameBoy (Color/Advance): VBA-M is a GB(C/A) emulator. It's a VBA fork, with multiple issues fixed and being able to run more games at higher speeds. Requires no special setup or BIOS.
Nintendo DS: NO$GBA (gaming version) is what you're looking for. Since GBA and NDS share some components, this emulator deals with both of them, and plays most DS games fine.
Nintendo 3DS: There's Citra, an experimental 3DS emulator. It's not ready yet, at all, but it starts to emulate first commercial games and some homebrew. Also, it's slow, bug- and crash-ridden, yet it's in heavy development, so don't expect to play 3DS games comfortably on your PC in 2015. But if you're waiting for it, just watch Citra's progress.
Sony PlayStation Portable: Recently, a really good emulator called PPSSPP appeared. It runs really well on most modern PCs and moderately powerful Android devices, so you can keep the "Portable" label with it. Go for the automated builds - they bring you the latest improvements, and since the developers are really active, you won't regret it. A lot of games are already fully playable, even on phones, despite being started around 2012. Doesn't require BIOS or any specific setup, just start and load your games.
Nintendo 64: Project64 pretty much owns this one. Beware, the installer contains a lot of adware. I've linked a page to CoolROM, which contains a packed, installer-free version - please use Alternative download link below the main Download Now to avoid hitting yet another downloader.
PS3/PS4/Xbox/Xbox 360/Xbox One: It's complicated. Short version: no emulators available for these platforms yet.
Long version: These platforms are too complicated and powerful to be emulated efficiently for now. There's RPCS3, PS3 emulator, and Xenia, an X360 emulator, but you can't really play any commercial games on these. Even homebrew fails to run properly on these, and aren't really usable yet. There are no emulators for PS4/X1 yet - don't touch anything that claims otherwise, as they're all virus-ridden fakes or donation buckets. Your computer (and bank account, passwords, Gmail and sanity, among other things) will thank you.
Geez, I don't want a separate emulator for every single console.
You're lazy. But that's okay.
RetroArch is a huge all-in-one emulator utilizing libretro and its cores (individual, lightweight emulators, systems or libraries like FFMPEG - you can use RetroArch as an emulator or a media player). It's available for many, many platforms (you can even run less power-demanding cores on Android, iOS, Blackberries, and even PS3, X360, original Xbox, Wii and GameCube) and supports a ton of cores.
Note that some cores require a BIOS or plugins to work properly. If a given game doesn't work out of box, you should check out the wiki and see if a given core requires some special setup.
I'm using OS X, and they're ugly, aren't available for Macs, and I want something better :'(
Why, OpenEmu to the rescue. It supports a huge amount of consoles (utilizing other emulators behind the scenes) with a well-looking, easy to use UI. It's quite big and requires a non-ancient machine with OS X 10.7 or newer. It supports a lot of consoles, but not golden DOS-powered oldies - Boxer, which is based on DOSBox, delivers you a nice UI for that.