Source Items are a great way of making your characters ultra powerful in FF7. You will pick up a few as you play through the game, but really making good use of them will require a lot of patience and clever use of the Morph Command.
There are 2 great places to find stat Sources in Final Fantasy VII. The first is the exploded reactor in Gongaga; here you can Morph Heavy Tanks for Power Sources. They're very easy to Morph in just one hit once your level is quite high.
The second place is the Sunken Gelnika. Each of the enemies here can be morphed into a different stat Source: Power Sources, Magic Sources, Guard Sources, Mind Sources, Speed Sources, & Luck Sources. These cover all 6 of Strength, Magic, Vitality, Spirit, Dexterity and Luck. You are advised to equip characters' ultimate Weapons while doing this; Yuffie's Conformer is particularly handy, and Barret's Missing Score can work well too with enough high AP Materia in it.
For a full list of locations where you can Morph for Sources, see the Morph List.
To use a Source, just enter the Item section of the menu, select the Source, and choose a character to use it on. The maximum a stat can reach is 255. However, you can give a character a fair few more Sources even once a stat is at 255 before the game won't let you give any more. This is probably because even with a stat at 255, it would drop below 255 with certain Materia equipped.
So a character with 255 Dexterity and a Mystile equipped has a Defence % of 113. Also note that the Luck stat affects the speed at which your time gauge fills up; not many people know this (I discovered it by raising Cloud's Luck to 255).
One further thing of note: with Magic at 255, Knights of Round will take off 9999 or very close to it with every hit, without an MP Turbo.