I know that people have different metrics for different stakes. From what I read it's usually 200x the big blind of the stakes you're playing at. I'm about to start saving again to build a Proper bankroll and starting using BR Management this time around to avoid going bust. So what is the proper bankroll rule to follow when saving/starting a bankroll?
First off, 200 times the big blind is NEVER* enough. That's two buy-ins. If that were your bankroll, you would go broke, even if you were the best poker player on the planet.
SoboLAN's answer was a great one for MTTs. For SNGs, you can probably start out a bit lower than that - around 60% of his recommendation should be ok if you are playing a relatively low number of games simultaneously and/or you are playing regular-speed games. If you are multi-tabling turbo SNGs, you should probably stick with the values he gave.
For cash games, Toby's answer listed a lot of the factors that you should consider. I'll add to that with some specific recommendations. If you're willing to deposit more money into your BR should you encounter a bad run AND/OR you are willing to move down in stakes, then you should be ok to use 25 buy-in base. As you move up in stakes, your edge on the field shrinks while the money becomes harder to replace and more meaningful. That should, generally, cause you to require a larger bankroll. In mid-stakes, a 40 buy-in minimum is pretty reasonable. Some prefer more. Then as you move into high-stakes, you should increase further. I have pretty limited experience with any stakes that would be called "high" (e.g. over $1k buy-in cash games), but when I was flirting with those games, I wanted at least 50 buy-ins.
*As another posted noted in the comments on the question, the one exception to my statement that 200 times the big blind is never enough is if you are playing fixed-limit games. In limit, it is proper to use a multiple of the blind to determine your BR requirements, and 200x is fine for starting out. As you progress to higher stakes, you'll want to move to 300x.
Deciding how comfortable you are with going bust (a.k.a. Risk of Ruin or RoR), is arguably the most important issue regarding how you personally manage your bankroll.
Issues like the depth of your bankroll, how you handle variance, the type of games you play, and potentially your drawdown (if you're living off the income from playing) are part of understanding RoR.
Check out this answer to find some useful links to online tools that will help you.
Since I rarely play cash games, I'll only talk about tournaments.
When it comes to tournaments, the bankroll you choose must depend on both your skill level and on your financial possibilies. You may be a millionare, but starting to play tournaments that have $5000 as buy-in is definitely not a good idea.
Basically, my rule is at least 60 buy-ins for tournaments. For example, if you have a bankroll of $100, then entering tournaments with $5 buy-in is not so good, but entering the ones with $1 is good.
If you start to increase your bankroll to a level where entering more expensive tournaments may be possible, then you can start playing such tournaments a little earlier. Maybe just to get a feeling about them. This bankroll management rule is meant to be bent (but not broken). Once you get your bankroll to about 150 buy-ins, you can take a step up the ladder more permanently.
The biggest problem when it comes to this is that poker players have a big ego and forget that you can't just climb up the ladder. Sometimes you have to climb DOWN. Some situations may come when you're getting low on cash and entering the usual tournaments doesn't make much sense financially. To use the previous example: if you get as low as $40, then entering $1 tournaments is a little too much, you may have to go down to $0.50 ones.
Most poker player's take this as taking a step backwards in their carrer. And they hate it. And they avoid it. And they get broke because of it. THEY'RE WRONG. This is not going backwards, this is simply adapting to the new conditions. When sitting at a table, you always adapt your play, taking into account dozens of factors (table image, stack sizes, field size etc. etc. etc.). You learn to be adaptable in your play, so why not when it comes to your bankroll ?
Bottom line: Those numbers that I gave you (60 and 150) are just guidelines. Make your own (reasonable) rules and stick to them. They're sacred rules. Bend them when you're at one of their edges, but never break them. If you do break them, you'll regret it. I PROMISE YOU.