Campaigning is the term for how you race your horse over a broad spectrum of time. An overall campaign would look at a horse's entire racing career. A campaign for a Trinity race might only look at 2-3 races leading up to the Trinity.
Knowing how to campaign your horse is going to help you prepare them for the more prestigious races and awards. One major factor in how you campaign your horse is their athleticism. A horse with low athleticism does not need to be raced as often, while a horse with average or high athleticism can probably race each and every game month
The way your horse performs in every single race they enter has some relation to their recent campaign. Racing horses the right amount keeps them focused and motivated. If your horse has 100% energy, they should always enter a race that month (unless you are sitting them out for other reasons). If they do not, they will lose morale, which will hurt their race performance.
Campaigns have an even bigger influence on your horses in races with $400,000 or greater purses - Auto-qualifiers, Trinities, Gemstone races, AGCs/EGCs, WCFs and League Of Legends. In these races, the standard campaign influence is there, but there is also a factor that looks at how the horse has prepared for the specific race. If you've heard the term "prep race" used in reference to real life racing, this is what this is looking at. The better a horse was prepared for these big races, the better they will perform here. Horses that are well prepared are usually going to run phenomenally in these races because they have tougher competition than they may see in other races, so don't be surprised if your horse runs some really stunning PRs.
April B, May A, May B - Trinities(9-12f, 3yos)
June B, July A, July B, August A - Crusader's Challenge (14-16f, 4yo+)
August B - Gemstones (4f, 2yos)
September A - League Of Legends (sprint/mile/classic, 4yo+)
October A - AGC/EGC (sprint/mile/classic/MA/SC)
November - WCF (sprint/mile/classic/MA/SC)
For more detailed information on special races, see here.
Preparing For a Big Race
In order to prep for a big race, you should buddy work and enter high graded races as much as possible to build experience. The more experience your horse have the more your horse will perform well in the big races listed above.