Total potential, or TP as it is commonly referred, has a very important part on Alacrity. With trialling, and more importantly in breeding, which is what I'm going to be discussing today.
There are 9 stats on your dog based off of real agility equipment found on a course:
Weave pole skill, A-frame skill, Teeter totter skill, Tunnel skill, Chute skill, Jumps skill, Tire skill, Dog walk skill, and Pause table skill
Your dog's TP is a sum of ALL 9 individual skill stats of your dog. Varies from each dog, even litter mates having different TPs. When you train your dog, a random number of points increase your dog's "Training Completed" stat. The closer it is to 100%, the better your dog will do in trials. There is also the chance that you can over train a skill; in which case the dog will have over 100% training completed; this is also a good thing and is normal. :3 When you have finished training a stat to or over it's fullest capacity, the stat will no longer be a link and will be BOLDED text. The term for a bolded stat is maxed because if you train by refreshing the page, a red alert bar will pop up and reads "This dog is already maxed in that stat!" hence the common usage of such a word. Maxed is also the term for a dog with all 9 stats bolded out. Please take note on the difference between "Maxed dog" and "Maxed "stat/s" when buying a dog.
Now that we know about TP and what it means; lets move on to more breeding information with the TP formula for litter estimates.

The TP formula is meant to give an estimate of the TP of your future litter's TP. When done properly, it is very helpful and often incredibly accurate.
(Mom's TP + Dad's TP) / 2 + 5 for every maxed (Or +90 if Mom and Dad are both maxed out) +/- 1, 2, or 3 points per stat (A 27 point leeway)
Feel free to copy and paste that to your profile page if you're afraid you will forget it.
Now let me take this moment to clear up several misconceptions:

When doing the formula, the /2 goes BEFORE the +5 per or +90.
If you put it in any order other than what has been stated, it might be incredibly inaccurate.

The formula only is accurate for the TP training completed.
You can have a 300TP dog and breed to a 600TP bitch and will end up with a >10 TP pup if they weren't started with training. How would it get 10 if 0+0/2 still makes 0? That 27 point leeway saved it from being 0.

Don't forget that little bit of training OVER 100% completed!
That small amount may not seem like much, but there's a HUGE difference from 304 at 100% and 304 at 105.09% (Which is actually 319.47TP) when being put into the game's own calculator that includes that itty little bit of TP difference.

There are other factors that can effect the accuracy of the TP formula. There are companions and account perks that add boosts; they will be discussed later. :3

These are often the most neglected stats on a dog, in my opinion. The speed, drive, and personality of a dog has a great effect on how well it does in a trial. Some players take pride in producing dogs with not only high TP, but high speed and drive as well. I personally also have an affinity for producing perfect temperaments in my lines.

From what I have gathered, perfect personality X perfect personality = continuing personality perfection. I have yet to see otherwise in that. How ever if even one personality trait has room for improvement, it can and usually will throw the entire litter's personality off.
For speed and drive, the formula to estimate what it would be in a litter is:
(Mom + Dad )/2 (Thank you Kaelizilla and Blix XD)
I feel so silly for not having heard of it until now; especially after playing for a year!

This is what makes your dog your dog. The massive numbers of patterns and opacities available to each dog on Alacrity, limitless possibilities on designs for dogs. That reach goes even further when you include "Glitch" opacities.

A "glitch" is a dog with any number of patterns that have an opacity setting over 100 or under 0. The unorthodox method of putting the setting too high or too low usually warrants a fascinatingly bizarre range of colors, every color imaginable. The tricky part is finding the correct base coat and pattern combination to get the color you desire. Once you learn intricate patterns involving glitches, you are probably going to be called a custom tool master.

On a dog it is possible to have up to 10 secondary markings, a white marking and a tan marking. The more markings the dog has, usually means it's more likely to pass on several markings on to it's offspring. There isn't a guarantee that all pups in a 12 for 12 breeding will have 12 markings, but there is that chance.
Another thing to mention is patterns in the same place. If you have a pair of corgi with black merle at Secondary #1 breed, there's an increased chance of Secondary #1 on all of the pups being Black Merle. An example of such an occurrence is my own corgi, Rouge. Her mom, Cahira and her dad, Italy are very closely matched in patterns. The entire litter was born looking exactly like mom and dad.

If you haven't seen it yet, the little badge on player's pages looks like this:

It is an account perk that costs 10 bones; it's always readily available for purchase on the bone exchange page. Now, I'm going to go a little in depth on the perks you recieve from it.

Guaranteed 4 in a litter: This is pretty clear XD In every breeding you produce, you will get 4 puppies. This is often very helpful, especially since a bitch has to wait 12 months (approximately 12 real days) before breeding again. If you know a pairing has wonderful pups and everyone is hounding you (no pun intended) to get a pup, having the knowledge that you will get 4 in a litter helps you plan ahead. You can decide then and there BEFORE the breeding if you're going to reserve any pups. It's often better than getting a 2-pup litter and having 4 people wanting one.
+10% TP boost: This little additive was often a savior for low TP breeding I've done to rescue a beautiful dog's lines. 10% may not sound like much, but when the TPs get in the upwards of 6-9k, 10% can often be too much (+600-900 TP). When you add a second account, the benefit doubles; giving the entire litter +20%. That little bit more can often make a litter excel from 600TP to 700TP

At the moment, there are only a few breeding companions available. They offer helpful features to make every breeding more meaningful.
Chance companions: These were given away as donation prizes for the Chance drive. They raise the TP of a litter by 15% and the effect is doubled when a Chance is equipped to both parents. It only has an effect PRIOR TO conception of the litter. You need to equip before your bitch gets pregnant or it won't work.
Easter Bunny: This companion was given away during our Easter Staff Egg hunt. It guarantees 4 pups in a litter when equipped to the stud or bitch. The effects do not stack.
Rabbit Companions: These companions were available during February 2011 for the Chinese New Year. The come in several varieties and offer the same effect as the Easter Bunny; giving a guaranteed 4 pups in a litter. The effects do not stack.

There are only 2 breeding items I can think of off the top of my head XP

Heat cookie: This item forces your bitch into heat at the next half hour and keeps her there until you remove it. This is a multiuse item so asking to borrow one is fine so long as you ask in sales chat
Chocolate Bunny: This item was available during our Easter Staff Egg hunt. This item pushes your bitch into heat immediately but is a single use item; so asking to borrow one isn't possible.

As more breeding- oriented companions are created and released to Alacrity, I will update this list with the items :3

I'm sad to say this seems to have turned into a war between us. With the massive influx of TP monsters, many are finding it difficult to remain in their niche.
TP breeders: These players breed for "brains" as they say. High TP is what they strive for, aiming for the Top Dawg list. Their route to sucess is often paved in months of training, selective breeding, the choice of who's not good enough to make it to be Top Dawg and dedication to producing the highest quality TP pups on the market.
Design breeders: These players breed for looks. TP doesn't have any place here; you're either beautiful and thrive or homely and set aside. They often aim for unique combinations and patterns. Their route to success is paved in conscientious breedings to beautiful dogs mostly descended from customs, the choice of who's not good looking enough to move on, and the dedication to move on even when their markets are down and out.
Cake Breeders: These players don't sacrifice anything. They're aim is high TP and beautiful dogs. I personally call this clique the "Cake" group because they have their cake (Beautiful dog) and can eat it too (High TP). Their road is paved in decidation, selective breeding and often leaves dozens of custom dogs in the wake that marked their lines' foundations.

This is all the miscellaneous stuff that really didn't belong in the other sections XD

Puppy Slots: When you have a litter born, you don't need a slot for every individual puppy. For the first 2 months (approximately 2 real days) of their life, all puppies share a single slot. This is often helpful when you breed several bitches in the same time frame and end up with 8-20 ish pups on your hands. The only thing is, after they hit 2 months, they need their own slots and dogs will run away if you don't have enough.
Line breeding / Inbreeding: ~filling in later XP
Out Crossing: ~filling in later XP
Up-breeding: ~filling in later XP
Breeding Partners: ~filling in later XP

Before you breed!!
Check your dog's lines: ~filling in later XP
Selecting A Proper Stud: ~filling in later XP

If you have a question, feel free to ask in the issue hub and message me or post here the question and answer for future readers! :D
Q: What is the age when I can breed my dog?
A: Studs/Males can breed at 8 months old; Bitches/ females can breed at 12 months on her first heat and every 6 months after.
Q: When is my dog no longer able to breed?
A: Dogs on Alacrity can breed until their dying day if their health permits
Q: At what age will my bitch go into heat?
A: At 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126; add 6 months.