Mephisto is one of the three prime evils, residing in the Durance of Hate level three, and is the Boss of act three. That's a lot of threes -- three to be precise! Mephisto is one of the more popular bosses to kill repeatedly in Diablo 2, likely because of his reasonably high drop rate of good items, and (potentially) short distance between his waypoint and the boss battle. If you can deal with loads of lightning, then get ready to face Mephisto.
Drop a town portal just after coming down the stairs to Durance level three. This will give you a safe fallback point if you die.
Clear the rest of the level up to the tops of the two side rooms. After crossing into the center of the level, you will see paths extending left and right. Each leads to a side room with a council member and other mobs. Kill everything in the room, and go to the north for a treasure or armor stand, but do not progress to the top center until you have cleared both sides.
Walk to the center of the top part of the level until you hear Mephisto. Do not walk further north in the center, or you will end up angering the blood lords behind Mephisto, who are better left alone until later.
Lure Mephisto back to the lava river separating the bottom and top of the level. Once he is at the top edge of the river, you can teleport or run around to the bottom edge and face him across the gap.
Attack Mephisto at range across the gap. Although this will take a long time, the AI may not let him respond, so you can kill him safely.
Skip the moat trick if you are not worried about dying and/or don't have the patience for a drawn-out battle. Instead, close with Mephisto and attack as swiftly as possible.
Attack with physical attacks or cold attacks. As you might expect, lightning attacks will not work very well against Mephisto.
Wear lots of lightning resist gear, as this will help you avoid taking too much damage.
Avoid the ice ball that Mephisto throws. He may be primarily a lightning demon, but the ice ball does the most damage.