I ran across this information on reddit a few years ago and the other thread got me remembering. Could be slightly out of date, so if you have any better recommendations please add below.

NES:

Mesen. Runs great and is very accurate.

https://www.mesen.ca/#Downloads

SNES:

Higan, for super accuracy.

https://byuu.org/emulation/higan/

SNES9x, for accessibility/features.

https://sites.google.com/site/bearoso/

http://www.s9x-w32.de/dl/testbuilds/?C=M;O=D

N64:

Mupen64Plus, for accuracy.

https://bitbucket.org/ecsv/mupen64plus-mxe-daily/overview

Project64, for a better interface.

www.emucr.com/search/label/Project64?&max-results=12

Gamecube/Wii:

Dolphin. Just great. Use the nightlies, they are way ahead of the latest stable release and usually work very well.

https://dolphin-emu.org/download/

Gameboy Advance (plus color/OG):

mGBA. Very decent. Also has an android core in RetroArch.

https://mgba.io/downloads.html

Nintendo DS:

DeSmuME. Not great, but for now the best on PC.

https://desmume.org/download/

DraStic. Paid app on Android, but a great emulator.

https://play.google.com/store/apps/details?id=com.dsemu.drastic&hl=en

Nintendo 3DS

Citra. Works well for some games, but still in development.

https://citra-emu.org/download/

WiiU:

Cemu. Works well for lots of games, but still has issues. Needs good hardware.

http://cemu.info/#download

Saturn:

Yabause, not perfect compatibility, but making good strides and has more features than other emulators.

http://emulation.gametechwiki.com/index.php/Yabause

Genesis:

BlastEM. The best for accuracy.

https://www.retrodev.com/blastem/changes.html

Kega Fusion. Not perfect, but also supports a bunch of other Sega consoles.

https://www.carpeludum.com/kega-fusion/

Dreamcast:

DEmul, good for some games, but not very accurate.

http://demul.emulation64.com/downloads/

NullDC, Reicast and ReDream are three in development emulators. NullDC is probably the most promising and might run some games better than DEmul.

http://www.emucr.com/search/label/nullDC-optimized?&max-results=12

PS1:

PCSX-Reloaded PGXP. Generally runs well and has some nice features.

http://www.ngemu.com/threads/pcsxr-pgxp.186369/

PS2:

Not as good as Dolphin for the Wii, but a pretty mature emulator that runs most games well.

https://buildbot.orphis.net/pcsx2/

PS3:

RPCS3. Runs some games pretty well, others not at all, but overall moving in great stried. Needs powerful hardware.

https://rpcs3.net/download

PSP:

PPSSPP. Runs very well, even on Android.

https://buildbot.orphis.net/ppsspp/

Arcade:

MAME. It's the go to for arcade stuff, there's an archive of about 60 gigabytes with a **** load of games.

http://www.mamedev.org/release.html

https://archive.org/download/MAME_0.197_ROMs_merged

https://journalxtra.com/gaming/download-complete-sets-of-mess-and-mame-roms/

(You also have to get them up to date to the current 0.198)

This is a pretty good resource for emulators:

http://emulation.gametechwiki.com/index.php/Main_Page

Good timing on this thread. I was just staring to look into this for my PC and Shield Pro.

If you're on a Mac, OpenEmu is a fantastic app. Good clean interface for your ROMs and you can download the emulator cores and go from there. You need the experimental version for MAME and it's very finicky with it, but otherwise it's awesome.
www.openemu.org

How do you actually get the games to play. I mean the old ones were on a disk or cartridge.

They ROMS can be found on the internet fairly easily.