String Slippage - Step one is to insure you are properly winding your strings around the tuners post so they do not slip and loose tension. Don't overlook this, slipping strings are the number one cause of tuning problems. Read more about that here. Locking tuners are also very helpful on electric guitars when using a tremolo.
 

Nut slots - When the slots in the nut are too small for the string gauge you are using the string can easily bind and catch in the slot.  When tuning, do you hear a repetitive creaking/popping sound? The nut slot may need to be cut larger to avoid binding the string...a simple an inexpensive procedure. You may also find it helpful to lub the nut slots to prevent binding. More about nuts.
 

Bridge Pins -When installing strings on an acoustic guitar it is important that the ball end be firmly seated against the bridge pin and bridge plate. If it isn't, the string may eventually pull upward, seat itself against the bridge pin and cause the tuning to go flat from the loss of tension.   Read more about changing strings here.
 

New Strings - Are they brand new strings or worse...nylon strings? New strings must have the opportunity to stretch. When replacing strings I recommend pulling on them lightly to "pre-stretch" the string. A fresh set of strings will probably require several re-tunes during their first day of play before they settle in and nylon strings...well patience is a virtue.
 

Worn tuners -Cheap or worn tuners often have slack in the gearing, the best way tune an instrument is to tune UP to the note. If you end up tuning a note too sharp, come back below the note and then tune up to it again thereby taking up the slack/play in the gears. You can also tug on the string that is only slightly sharp to bring it's pitch down. Poor quality tuners with low gear ratios make precise tuning more difficult as the shaft rotates a greater distance with even the slightest turn of the knob. (Just so we're clear... a 21:1 tuner is far superior to a 11:1 tuner as it takes 21 complete revolutions of the tuners knob to turn the string post 360°.)
 

Tremolo - Does your electric guitar have a tremolo? And you're still wondering why it goes out of tune? :)
Every time you use a tremolo it must return to the absolute same position it started from to maintain perfect tuning. Heavy tremolo use stretches the strings and will eventually cause a loss of tuning even if one has the strings clamped downed securely. And if you are using a Floyd Rose style tremolo, tension loss (or a broken string) will cause all strings to loose their pitch. All pivot points should be lubricated to reduce friction and allow the tremolo to move with ease. Read more about tremolos here.
 

More serious problems There are other problems (structural problems) that can cause tuning problems as well. Loose necks (they shift around once tension is applied), a loose and rising bridge, sloppy saddles flopping back and forth, a rubber neck with no stiffness and other structural problems.