Chip's Challenge 1 on 64-bit machines
Microsoft's Entertainment Pack version of Chip's Challenge, commonly abbreviated to MSCC, is a 16-bit program, which means it can run on 16-bit and 32-bit versions of Windows, but it will not run on 64-bit ones. However, most Windows computers sold today are 64-bit, which unfortunately means Chip's Challenge will not run on them. To get around this, many emulation clones of Chip's Challenge have been created, such as Tile World, but despite being able to look and play somewhat like MSCC, even to the point of emulating strange and obscure glitches like the Convergence Glitch, these clones are never completely accurate to what MSCC actually plays like. If you have a 32-bit version of Windows Vista or Windows 7/8/10, MSCC will run fine; but if you don't, here are a few solutions to the problem.
Andrew R. discovered a cool tool that allows you to run old 16-bit games like Chip's Challenge on 64-bit versions of Windows, called WineVDM. You can download it here, with some more information about it here. Andrew has some more information and instructions for installing it here. This tool has received highly active development; it appears that all of the bugs and quirks that were discovered with it so far have now been fixed.
A script that automatically installs and sets up the program so you just have to open CHIPS.EXE as normal created by Sickly Silver Moon can be found here
If you have Windows 7 Professional or higher, download Windows Virtual PC and XP Mode, a free download from Microsoft's website. This will allow you to run a 32-bit version of Windows XP on your Windows 7 computer, which will allow you to run CC while running the virtual PC.
If CC doesn't seem to play as smoothly as usual, using VMware instead of Windows Virtual PC to run Windows XP Mode may seem to run CC better (turning off CC's sounds may help too). Here is an article on how to set this up.
Note that Windows XP Mode is not available on Windows 8, and is not recommended anymore for Windows 7 due to Windows XP support being discontinued. However, if you still have it on a Windows 7 64-bit PC, then this option will most likely work for you.
Get an old 32-bit machine
A similar (although more expensive) option would be to purchase a copy of 32-bit Windows and install it as a virtual machine on your PC - or alternatively, you can go to a shopping location of your choice that sells old computers. Buying a physical computer may be the less expensive option, as some can go for as low as $35 including shipping. Of course, a computer of that age does have its quirks, but for playing CC it should work just fine.
If you have or can get a copy of Windows 3.1, you can use a program called DOSBox which can run CC through Windows 3.1. Here is a guide by ChipHome5 on how to set this up. Note that this option may be the most difficult to set up.
Mac and Linux
Users of Mac and Linux operating systems can run MSCC using the Wine compatibility layer. This should work regardless of whether the system is 32-bit or 64-bit.