Grades of Races
There are a variety of "grades" in which you can enter your horses. Each grade caters to a different level of talent, with the lowest level being claiming (or maiden) races and the highest level being G1 races. You should make every effort to enter your horse in the grade which they are suited for. Repeatedly downgrading your horses (entering them lower than they should be) is against the rules and is considered cheating. It is unfair to the horses who actually belong in the lower grades to have substantially better horses in their field.
The pages discussing Peer Comparisons and Performance Ratings will help you figure out what grade your horse belongs in. Keep in mind that PRs generally start out low and improve when a horse is at their peak. Peer Comparisons are the most accurate depiction of how a horse will race at the present time and are the best gauge of what grade to enter regardless of past performance. Ex: A 2 year old might run 60 PR when they are not peaked, but by the time they turn 3 they could be capable of running 160 PR and should be entered in G1 at that point.
Claiming: The lowest grade of races. Entering these may place your horse for "claim", where other stables can purchase them from the race. You can also view horses in these races to claim for your own stable. (Currently, claiming races do not allow claiming, so it is safe to enter there.)
Maiden: These are races for horses who have never won a race. You will mostly see these for two year olds, but they may show up for older horses occasionally.
Allowance: Allowance races are the bread and butter of racing. They'll make up the largest portion of all the grades, and will usually be the fullest. The horses in these races are toward the lower end of the talent spectrum, but having consistently placing allowance horses can still bring in a steady income.
Allowance NW_: These are allowance races that are actually designed for horses moving up from lower ranked races. NW stands for Not Won (or No Wins), and the number following determines the number of races a horse can't have won. For example, a NW2 race means that entrants cannot have won 2 races (or cannot exceed 1 win). NW3 races are only open to horses who have not won 3 races (or cannot exceed 2 wins).
Listed Stakes: Listed stakes (ungraded stakes) are races listed with a name, but no grade indicator (G_). These are the lowest level of stakes, which are meant to give you a taste of racing in stakes. Horses who race here are usually better than allowance level horses, but aren't good enough for graded stakes. Some of them may be younger or less developed, and could mature to race in stakes races later in life. Ungraded stakes do not give points.
Graded Stakes: Graded stakes are races which have a name followed by a grade: (G_). All graded stakes races award points to the winner(s), but the amount of points given varies by grade. G3, or Grade 3, is the lowest level of graded stakes. It is a small step up from ungraded stakes, and is a good place to start earning more money or qualifying colts to stud. G2, or Grade 2, is the middle level of graded stakes. These are held less often than G3 races, but more often than G1 races. G1, or Grade 1, races are the highest level of races on the game; therefore they are also typically the most competitive. G1 races also award the highest amount of points, and the highest purses.