The Secret of Monkey Island
This page is a growing index of various tools & resources for DOS gaming in the modern era.
DOSBox is *the* must-have tool for playing DOS games on modern hardware. It's an open-source emulator of both DOS itself as well as the hardware on which it was typically run back in the day. It's available for lots of different platforms and has great compatibility with many games.
DOSBox has lots of configuration options and built-in tools to help you set it up just right. Explanations of such and other things can be found on the DOSBox Wiki.
If you're using Windows, there is a nice graphical front-end for DOSBox available, called D-Fend Reloaded. It might also be interesting to check out DOSBox Enhanced Community Edition, which adds some new features and bugfixes to DOSBox which aren't included in the official release, such as 3Dfx support.
If you're using Mac OS X, there's a user-friendlier app for running DOS games called Boxer. It uses DOSBox internally, but provides a nicer UI and workflow.
Play DOS games in your browser
Started back in 2013, playdosgames.com is one of the first sites to offer DOS games playable in your browser. The games are added & tested by hand and also offer an explanation of how to play. The site only hosts real abandonware and otherwise demos & shareware, so everything offered here is completely legal.
MS-DOS GAMES ARCHIVE
The awesome folks over at Archive have a huge collection of DOS games on their website. The games seem to be added in bulk, without much manual testing, so your mileage may vary. But the amount of games they have on offer is staggering!