I can relate. I like to play my guitar late at night as well. It helps me unwind and relax.
The least expensive way to accomplish your goal is a small headphone guitar amp such as the Vox amPlug AC30 that looks like the one pictured below which you can find for under $20.00:
Or if you want more controls - a Concept 3 Guitar Headphone Amp is less than $25.00 and gives you an auxiliary input for your i-pad or other device and a several controls that allow you to tailor your sound.
Both of these are small enough to fit in your gig bag and run on batteries and are very easy to just plug in and play.
Other alternatives have been discussed in other answers and include effects pedals with headphone output as well as one of the many newer guitar amplifiers that offer a headphone jack for silent practice.
Have fun rockin out in the wee hours while the rest of the family sleeps!