The answer to how is lightning formed is easy to understand if you know that lightning is a result of several things happening very quickly. There is a whole "behind the scenes" set up to make that bright bolt of lightning we see in a stormy sky.
Clouds become charged, sending out leaders towards the earth. The earth being the opposite charge begins to send out streamers of its own toward the clouds. When these two meet it creates a channel, acting like a metal rod connecting the sky to the earth. Lightning is natures way to equalize the inbalance between the clouds and the earth. The lighting strike is the massive current flowing through this completed channel.
All this leads to the other part of lightning we all recognize - the flash. Believe it or not, air can explode. That massive current flowing through the completed channel is extremely hot, hotter than the surface of the sun actually. This extreme heat causes the air to rapidly expand creating the flash we are used to seeing. These steps also lead to the formation of thunder; lightning's partner in crime. This exploding air starts a compression wave that becomes the rumbling thunder we associate with an active storm cell.
If you would like a detailed explanation of the formation of lightning and thunder follow this link.