Bill, you'll be happy to know that just yesterday when I taught a beginners self defense class, I took the time at the end of class to define and demonstrate simple terms and concepts, one of them being "base."
The class started with the technical stand-up, which I told them is also called "standing up in base." I realized more white belts don't get what "base" means so I explained it explicitly.
I had them do a very simple drill to understand their base and balance. They just stood with their arms up in a basic defensive stance, and their partner walked around pushing and pull on them. They have to try to sink their hips and adjust their feet to keep a strong base. They lose when they are tipped over, usually when their head or hips end up in front of their toes or behind their heels.
That's a very, very simple drill, and boring to anyone with any experience, but I've found it to be a good for totally green white belts like I was teaching.
There are ways to stand up without base, like trying to hop straight to your feet and stand up with your feet together, which I had them try while their partner pushed on them. They kept stumbling and falling back down until they did the correct "stand up in base."
They were also learning the basic Gracie Jiu-Jitsu 101 clinch which has a strong emphasis on "base." You can almost hear Rorion say "base" every time you drill these because he drove that point so hard in the old VHS tapes.
From the clinch, I'd have their partner try to bully them, trip them up, and toss them around. They had to find how to position themselves so they weren't easily affected or off-balanced, while also being able to pressure and control their partner.
I define base as how you position yourself to maintain your balance and connect yourself to the ground. Your base is often on your feet or knees, but it can use your hands, elbows, shoulders, or head.
Your base can be connected to your opponent, like posting a hand on their hips, or your chest against their chest. Having your base on your opponent also be dangerous because then their motion can disrupt it, but this connection is also what allows you to apply pressure and control them.
A solid base tends to be wide with a low center of gravity. That center is usually in your hips, which is why sinking and dropping them down is so emphasized.
Good sweeps attack the opponent's base to off-balance them. If you can't break their base, then the sweep will be difficult and require too much effort and strength.
The words "base" and "balance" are often used interchangeably, but to me, balance is a slightly different concept.
The dictionary defines balance as "stability produced by even distribution of weight on each side of the vertical axis" and "the ability to retain one's balance."
The dictionary doesn't really have a definition for base as we use it, though the closest one is "the bottom of something considered as its support," which is comes from the word's use in architecture (like the base of a building or column).
I define balance is the ability to use one's sense of gravity to stay on top and maintain position.
Of course, that's usually done by maintaining a good base, but I believe you can have balance in moments where you lack base, like being able to ride sweeps or throw without being flipped. If you break it down in slow-mo, this balance is usually because they maintain a sense of their base while in the air, and they adjust themselves so they land in base, or in a way that lets them quickly regain their base.
What we mean by "good base" also factors in the the context (like a self defense situation) or rules of our sport. You see wrestlers achieve incredibly base by going bellydown and throwing their arms and legs out like starfish to prevent themselves from being flipped over and scored against. That will get you destroyed anywhere else, but hey, great base!
As people are already bringing up, there are different approaches to mainting base and balance (active, dynamic, static, slow, fast, etc.), and how posture ties into all this.
I'll try to get Sloth (Josh Vogel) in on this since he has an instructional series through
www.bjjworkouts.com that focuses on very core concepts of base, balance, and posture.
