Trainer points are what help you to excel.
Basically, the more points you have, the higher you Trainer Level.
The higher your Trainer level, the more Pokémon you're allowed to have.
Simple as that.
Obviously, if you win battles, you get more points than would if you lost them.
Here's the list:
Single Battle: Win=3 Lose =1
Double Battle: Win=9 Lose=3
Tag Team Battle: Win=13 Lose=5
Here's the level up guide:
To level from level 1 to level 2, you need to have earned 20 Trainer Points.
To level from level 2 to level 3, you need to have earned 30 Trainer Points.
To level from level 3 to level 4, you need to have earned 40 Trainer Points.
To level from level 4 to level 5, you need to have earned 50 Trainer Points.
To level from level 5 to level 6, you need to have earned 60 Trainer Points.
To level from level 6 to level 7, you need to have earned 70 Trainer Points.
To level from level 7 to level 8, you need to have earned 80 Trainer Points.
To level from level 8 to level 9, you need to have earned 90 Trainer Points.
To level from level 9 to level 10, you need to have earned 100 Trainer Points.
If your character is under 16, he/she cannot go above level 8.
If your character is under 13, he/she cannot go above level 5.
Students: Trainer Levels 1-8
Teacher Assistants: Trainer Level 9
Teachers: Trainer Level 10
Like I said earlier, the higher your Trainer level, the more Pokémon you're allowed to have.
Here's the guide on how many Pokémon you're allowed to have depending upon your level (Thanks to Narakun for thinking this part up):
Trainer Lv. 1: 3 Pokemon at a time--all must be basic
Trainer Lv. 2: 3 Pokemon at a time--1 stage1 Pokemon
Trainer Lv. 3: 3 Pokemon at a time--2 stage1 Pokemon
Trainer Lv. 4: 3 Pokemon at a time--3 stage1 Pokemon
Trainer Lv. 5: 3 Pokemon at a time--3 stage1 Pokemon or 1 stage2 Pokemon
Trainer Lv. 6: 4 Pokemon at a time--3 stage1 Pokemon or 1 stage2 Pokemon
Trainer Lv. 7: 5 Pokemon at a time--4 stage1 Pokemon or 2 stage2 Pokemon
Trainer Lv. 8: 5 Pokemon at a time--5 stage1 Pokemon or 3 stage2 Pokemon
Trainer Lv. 9: 6 Pokemon at a time--6 stage1 Pokemon or 3 stage2 Pokemon; may assist teachers
Trainer Lv. 10: 6 Pokemon at a time--any stage Pokemon; teach classes
And there you have it.
If you're confused about anything, please PM me.
-Aima-chan