When jungling, why do certain champions get blue first or red first?

Solution 1:

Usually champions that rely heavily on their mana or abilities to jungle get Blue first, so that they can use abilities without worrying about mana consumption or to lessen their coolodwns by 20%. Also, the Blue buff gives enough xp to go from level 1 to 2 quickly.

Champions that gank really early, such as Shaco, take Red buff first to slow down the enemy and deal extra damage. Another reason for starting at red, is that some champions do not require mana for their abilities, for example Lee Sin and Shyvana.

Solution 2:

If you want to pull off a level 2 gank with any chance of success you need the red buff. If done properly you can gank lane at level 2 while enemy is still level 1, meaning better base stats, more abilities and a slow from red buff that can be applied over and over again till they die.

Blue buff is the typical jungle start if you aren't going to level 2 gank.

Solution 3:

Blue buff starts are for champions that are either mana dependent or cannot gank effectively at low levels to warrant starting at red. Some champions also start blue in order to steal the enemy jungle's red buff due to a faster jungle speed. Popular champions that do this are Shyvana and Udyr.

Red buff starts are for champions that have high gank potential and/or do not utilize blue buff effectively. Oftentimes, a red buff jungler will gank at lvl 2, or go pass blue buff to their AP mid early.