Basics of Numpad Notation
Numpad notation is a system for writing the directional commands used in fighting games in an easy-to-read, easy-to-understand way, even if it may not seem like it at first. The core mechanic of numpad notation is also how it gets its name: numbers are used in place of directions, according to their position on a keyboard's numpad.

Directional Commands
456 N
All directions are written assuming the character is facing right (1P side). This replaces the abbreviations or individual direction names used in other forms of notation. Additionally, it uses common abbrevations and notations to indicate how moves are to be done during combos.

An advantage of numpad notation is its ability to simply and consistently indicate complex movement sequences. You're not really saving any space when you compare QCF (Quarter-Circle Forward) to 236, but writing [2]8462 is easier than "hold down, up back forward down" no matter how else you might try to write it.
These are pretty self-explanatory, but just in case: when buttons follow a numerical movement command, they indicate the attack button to press.
For BlazBlue, these are A, B, C, and D. Guilty Gear uses P, K, S, H, and D. If an attack button is listed on its own, the direction is assumed to be 5 (Neutral).

Gatlings are attacks that cancel from one to the next, be it a normal move, a special move, or a super attack. In BlazBlue and Guilty Gear, most combos are built using gatlings. These are indicated by a > between moves. Note that the term gatling comes from Guilty Gear, and that in any official BlazBlue materials, they will be called Revolver Action. However, most everyone in the english speaking world uses the term Gatlings.
Example: 5A > 5B > 5C = Neutral A, cancel that attack's recovery with neutral B, then cancel that attack's recovery with neutral C.

Holding/Releasing Buttons and Directions
Sometimes an attack or command requires an input to be held, and then released later. Brackets are used for both, [] for hold and ][ for release. You'll see a lot of these in Carl combos.
Example: 5[D], 22]D[ = press D while neutral, then press Down twice and release D to perform the second attack. This same notation can be used to denote holding directionals. For example, [2]8D is the command for Jin's Astral Finish.

Immediate Gatlings/ Kara Cancels
Occasionally you'll need to press one button immediately after another to perform special attacks or to kara cancel (allowing you to use the movement of an attack that you cancel to reach farther with the attack you cancel into). These are written using a tilde (~) between the attack buttons.
Example: 225D~C = press down twice, return to neutral and press D, then immediately press C. This allows you to use the movement of 5D while actually performing 22C.

Links are attacks that do not cancel into each other, but that still combo thanks to hitstun. You must wait for the first attack to completely finish, then start the second attack after the first one ends. Links are written with commas between attacks.
Example: 5B, 5C = neutral B, then neutral C after the B attack recovers.

Sometimes you'll see a prefix before a numerical movement command or attack. Here are some of the most common:
j. sj. Jump and Superjump
j. indicates jumping or in midair. sj. indicates a superjump.
Example: j.2C = perform a Down+C attack while in midair.

c., f. Close and Far
Sometimes the same command can do 2 different attacks depending on your distance to the opponent. Close is usually right next to the opponent, while Far is everywhere else. While BlazBlue does not have any close/far attacks, this is a common occurance when playing other games like Guilty Gear.
Example: c.5A = press Neutral+A while close to the opponent

d. Command performed while in Drive (Noel's Chain Revolver moves are notated this way)
5D > d.6D = press neutral D to initiate Drive, then press forward D while in Drive for the next attack.

tk. Tiger Knee Indicates that the command should be performed on the ground with an upward end to the motion, allowing it to be performed just as you enter a jump state.
Example: tk.236A = Down, Down-Forward, Forward, Up-Forward, then press A just as you enter the air. This may also be written as 2369A.

(Fatal) Counters and Rapid Cancels
Some combos only work under special conditions. One of the most common is a combo that's started with a counter hit - because counter hits have additional hitstun, they allow for combos that would otherwise be impossible. When an attack has to be a Counter Hit to work properly, (CH) is written after the attack. If it needs to be a Fatal Counter, (FC) is used instead. Furthermore, some moves don't link together naturally, and require a Rapid Cancel for the combo to continue. When a Rapid Cancel is used in a combo, it is abbreviated as (RC), and you'll use it to cancel the move that came before it.
Example: 5B (CH) > 6A > (RC) > 236B = neutral B hits as a counter, then cancel to forward A, and rapid cancel that into quarter-circle forward B. Note: unless links have an especially long wait on them, timing is usually not written. Typically the next move is performed as soon as it's available to use and will hit the opponent.
Jump Cancels
The name sort of gives it away. This is written as a Gatling into JC.
Example: 6A > jc > j.B > j.C > 236C = Forward + A, cancel with a jump and attack with neutral B, cancel with neutral C, cancel into the special move quarter-circle forward + C.

Less Common Notation
Canceling Multi-hit Moves Early
If you need to cancel an attack that hits multiple times before all of the hits connect, it's typically indicated by adding the number of hits desired in parentheses to the end of the attack.
Example: 41236C(1) = cancel after the first hit from half-circle forward C.

Some moves allow additional commands to be entered afterward to change how the first command works. Hazama demonstrates two of the most common forms: a stance, and a variable Drive. Followups are written using an arrow (->) between commands.
Example: D -> A = press neutral D, then press neutral A for the followup.

Multiple Possible Inputs
This usually isn't used in combos, but is often used in move lists. When multiple buttons can be pressed with a single movement input, slashes (/) are used to separate the possible inputs.
Example 1: D -> A/B/C/D = press neutral D, then you may press either A, B, C, or D.

Example 2: [4]6A = hold Back, press Forward + A.

Repeating Inputs
Some attacks require, or allow, an attack button to be pressed multiple times. It's simple math: Command (xNumber of times to perform the command). If an entire group of commands is to be repeated, the commands will all be surrounded in parentheses with the multiplier outside.
Example 1: j.C(x5) = while jumping, press neutral C five times.

Example 2: 5C(xN) = press neutral C forever, or until it stops working.

Example 3: (5A > 5B > 5C)xN = loop neutral A, neutral B, and neutral C forever.

Whiffs and Other Unusuaul Things
You can't have shorthand for everything. Sometimes an unusual circumstance comes up that can't be adequately explained in three letters, and so instead we simply give instructions in parentheses.
Example: BC (whiff) > 6C = press B and C together to perform a throw that misses the target, then cancel to forward C.

Example 2: 236236B, (adjust) 4D = double quarter-circle forward B, then adjust your position in order to make the back D hit.

Attack After Landing, Homing Jump (Guilty Gear Only)
Links can sometimes start with an air move and finish with a ground move. When this is the case, the gap between moves during which you land on the ground is indicated with |>. In Guilty Gear, combos performed in the superjump following a successful Dust attack are indicated with two opposing slashes - /\.
Example: j.5C |> 6C = perform a midair neutral C, then land and perform a forward C.

Example 2: 5D /\ j.P > j.K > j.S = perform a standing Dust attack, then jump to perform the Dust superjump and, while in midair, press punch, kick, then slash.

Move Names, Placeholders
In combos, you may find the name of an attack used in place of an input. Just perform the move that's written. Additionally, combos might end with phrases like "air combo ender," "OTG ender," things of that nature. In these cases you would use any general combo that suits the situation and yields the result you want, rather than writing out a new combo for each possibility.

Abbreviations and Acronyms
There are MANY abbreviations and acronyms that people use for the various occurances in BlazBlue. Here are a few of them.
AbbreviationStands ForExplanation
FDFaultless DefenseWhile in BlazBlue this term is actually called Barrier Block, most people use the term FD since it is almost the same thing from Guilty Gear.
IBInstant BlockSelf Explanatory
RCRapid CancelIn addition to that, this term can also refer to Roman Cancels in Guilty Gear, which are exactly the same thing.
JCJump CancelSelf Explanatory
CHCounter HitSelf Explanatory
FCFatal CounterSelf Explanatory
TKTiger KneeSee Tiger Knee in the Prefix Section
HJC/SJCHigh Jump Cancel/ Super Jump CancelThe two are snonymous with each other
IADInstant Air DashSelf Explanatory
OTGOff-The-GroundAttacks that hit the opponent when they are fallen and hit them Off The Ground.

Real World Examples
2[A]~6]A[ > 2A > (6A Bug), 5C > (C Bug), Taunt~D > (D Bug) > sj.2A~[C] > (A Bug) > j.]C[ > j.2B (whiff) > (C Bug), Taunt~AD > (D Bug) > (A Bug)

Down + Hold A, immediately release while pressing Forward, cancel into Down + A again, the Forward+A Bug hits, link a neutral C, let the C bug hit while you taunt (immediately press D while taunting), D bug hits. Perform a superjump and press Down+A to dive. Hold C while diving. A bug hits, let go of C, press Down+B in midair to do another dive that whiffs. C bug hits, taunt then press A and D. D bug hits, then A bug hits.

Already we're seeing the benefits.
5B > 2B > 6C > jc > j.D, 6A/2B > 623B, 5D -> air combo ender

Neutral B, cancel into Down B, cancel into Forward C, jump cancel into neutral D, land and perform either Forward C or Down B, cancel into Shoryuken+B, link neutral D and finish up with your preferred air combo.