Be aware that type disadvantages and advantages do change the effectiveness of the assists though not so much to make noneffective ones useless. (most of the time)
Fighting poke assist gives you two loops with just one. In short, it doubles your power.
Water type poke assist is amazing in this game. Trap anything in a bubble, then activate Fighting and make the capture.
Psychic levitates all Pokemon, and if used when they are close together, you can activate a quick fighting and get them all at once.
Dark sacrifices some of your styler energy to greatly increase the length. This is almost necessary to capture large Pokemon such as steelix
Electric recharges your styler energy depending on the level of assist (number of symbols next to assist)
Partner Assist This powerful electric assist only possible with your partner Pokemon minun (boy) or plusle (girl). In this, your partner shoots out beams of electricity that hit temporarily stun the Pokemon being captured. (except for ground types, who are unaffected) More segments of the partner energy bar filled means a larger hit range and longer paralysis. (more segments of the gauge are unlocked as the game goes on) This will be your most valuable assist, and you will learn to rely on it for longer captures. Also works very well with fighting.
Honorable mention-Ghost Whenever you make a loop with this assist on, it creates a ghost/shadow that traps the Pokemon being captured. As you would be looping almost the whole time in a capture, the Pokemon would stay trapped. This is an assist that is great when it works and is good in some situations. However, it is not as reliable as those listed above.
The others in this game can be useful, but are usually less reliable than these listed here. Some types have no assist at all. Obviously you can use water and psychic without fighting, but they are brilliant combos for those tough captures. Also, it is worth noting that dark and fighting can not be used together, as when one expires the effect wears off. Assists that affect the Pokemon being captured keep their effects temporarily, so Dark or Fighting can be used directly after for an easy capture. Levels other than electric affect only the assist time, not what they do. Also, another assist can be used when there is still time on the clock. This is how water/fighting can work, since if you waited for the clock to be up the bubble would be gone.