Pretty much every part has an identifying number cast or stamped (sometimes both) into it. Decoding all these numbers is a pretty involved topic that can, and does, take up entire books, so this will just be a small set of the most important information.

Vehicle Identifaction Number

Each car has a unique VIN. For our cars, it's located on a stamped steel tag on the front edge of the driver's door frame. Later cars have it on a tag rivited to the driver's side of the top edge of the dash. Unlike later VINs, our car's VINs don't tell you very much about the car. For early-mid 60s Mopars built in the US, the VINs have 10 digits arranged as LMYAssssss.
Digit1962196319641965
L Car line
1Plymouth Valiant Slant-6
2Plymouth Slant-6
3Plymouth V8
4Dodge Dart Slant-6
5Dodge Dart V8
6Dodge Custom 880 V8
7Dodge Lancer Slant-6
8Chrysler V8
9Imperial V8
1Plymouth Valiant Slant-6
2Plymouth Slant-6
3Plymouth V8
4Dodge Slant-6
5Dodge 880 V8
6Dodge V8
7Dodge Dart Slant-6
8Chrysler V8
9Imperial V8
1Plymouth Valiant Slant-6
2Plymouth Slant-6
3Plymouth V8
4Dodge Slant-6
5Dodge 880 V8
6Dodge V8
7Dodge Dart Slant-6
8Chrysler V8
9Imperial V8
LDodge Dart V8
VPlymouth Valiant/Barracuda V8
1Plymouth Valiant/Barracuda Slant-6
2Dodge Dart Slant-6
3Plymouth Slant-6
4Dodge Slant-6
CChrysler V8
DDodge Polara, 880, Monaco V-8
LDodge Dart V8
RPlymouth V8
VPlymouth Valiant/Barracuda V8
WDodge V8
YImperial V8
M Model
1Valiant V100, Savoy, Dart, 880, Lancer 170, Newport, Custom
2Belvedere, Dart 330, 300 Sport, Crown
3Valiant V200, Fury, Dart 440, Lancer 770, New Yorker, LeBaron
4Valiant Signet, Sport Fury, Polara 500, Lancer GT, 300H
5Valiant V100, Savoy, Dart, 880, Lancer 170, Town & Country Wagon
6Belvedere, Dart 330 Wagon
7Valiant V200, Fury, Dart 440, Lancer 770, New Yorker Town & Country Wagon
8Taxi
9Police
0Fleet Car
1Valiant V100, Savoy, 330, Custom 880, Dart 170, Newport, Custom
2Valiant V200, Belvedere, 440, 300 Sport, Crown
3Fury, Polara, Dart 270, New Yorker, LeBaron
4Valiant Signet, Sport Fury, Polara 500, Dart GT, 300J
5Valiant V100, Savoy, 330, 880, Dart 170, Newport Town & Country Wagon
6Valiant V200, Belvedere, 440 Wagon
7Fury, Polara, 270, New Yorker, Town & Country Wagon
8Taxi
9Policei
0Fleet, 880, 300 Pace Setter
1Valiant V100, Savoy, 330, 880, Dart 170, Newport, Custom
2Belvedere, 440, 880 Custom, 300 Sport, Crown
3Valiant V200, Fury, Polara, Dart 270, New Yorker, LeBaron
4Valiant Signet, Sport Fury, Polara 500, Dart GT, 300K, Barracuda
5Valiant V100, Savoy, 330, 170, Newport Town and Country Wagon
6Belvedere, 440, 880 Custom Wagon
7Valiant V200, Fury, 270, New Yorker Town & Country Wagon
8Taxi, New Yorker Salon
9Police
0Fleet
1Valiant V100, Belvedere I, Fury I, Dart 170, Coronet Deluxe, Polara, Custom 880, Newport
2Fury II, Coronet, 300 Sport, Crown
3Valiant V200, Belvedere II, Fury III, Dart 270, Coronet 440, Custom 880, New Yorker, LeBaron
4Valiant Signet, Satellite, Sport Fury, Dart GT, Coronet 500, Monaco, 300Ln
5Valiant V100, Belvedere, Fury I, Dart 170, Coronet Deluxe, Polara, Newport Town & Country Wagon
6Belvedere, Fury II Wagon
7Valiant V200, Belvedere II, Fury III, Dart 270, Coronet 440, Custom 880, New Yorker Town & Country Wagon
8Taxi, Barracuda
9Police
0Belvedere Superstock, Coronet Superstock
Y Model Year 1962 1963 1964 1965
A Assembly Plant
1Lynch Road, MI
2Hamtramck, MI
3Jefferson, MI
4Belvedere, MI
5Los Angeles, CA
6Newark, DE
7Saint Louis, MO
9Windsor, Ontario, Canada
ssssss Sequence number The 6-digit sequence number starts at 100001 for the first car of the model year at each assembly plant, and increses sequentially for each car on the assembly line

Casting Numbers

 Any decent treatment of casting numbers would likely fill more space than should go into a FAQ. Each part has its own location and list of casting numbers and stampings. The best thing to do would be to post the numbers you've found to the mailing list, and several helpful people will tell you what you have.

You can also try looking at this page, which has a lot of casting and stamping number identification information.

Engine Stampings

Slant 6

FIXME I don't know anyhting about where the stamped numbers are or what they mean

Small Block

The ID pad is a flat machined pad at the front driver's corner of the block, just below the head. It can be hard to read, sometimes obscured by the exhaust manifold, or years of oil and dirt caked on.

There dosen't seem to be a simple pattern to small block stampings, but you will see the engine's displacement, or the first two digits of it. There's a more complete treatment of what you might find here in Year One's technical info section.

Big Block

For the low-deck B engine, the ID pad is a flat machined pad on the passenger side near the distributor base below the head. On the RB block, it's located at the front of the block, between the water pump housing and intake manifold, and between the distributor and driver's side head.

The first line will contain one or two letters, followed by 2 or 3 numbers. The first letter tells you the year:
PRSTVA
196019611962196319641965
After 1964, the letter starts over at A for '65, B for '66, etc. Skipping I, so 1973 is J

The numbers denote the engine's displacement, or the first two numbers of it. For example, on a 413, you may see "41" or "413". For a 383, you might see "38" or "383". "A 38" would be a 1965 383.

The second line tells you what day the engine was assembled. You'll see either a month and date, or a 4-digit date on the 10,000 day calandar, followed by the sequence number for that day. "2 10 04" would be the 4th engine built on February 10th.

The third line contains special information to the original builder of the engine such as oversized bores or undersized crank.

Fender Tag

Complete information about deciding the fender tags on our cars would be too much to go into this FAQ, too. Each year was slightly different, and each factory was also different. There are some people on the list with all the information necessary to decode the tag; you just need to post the VIN (which gives the year and assembly factory), and the contents of the tag. Instead of listing it like this 
ABCDEFGHJKLMNPRSTVWXYZ
13  3   72   52    3 3
It's better to give it like this, since it's easier to read 
A=1, B=3, E=3, J=7, K=2 P=5, R=2, X=3, Z=3
The different sections on the fender tag are: 
SO NUMBER
Shipping order number		 Planned delivery date MDDM: month 1(Jan) - 9(Sep), A(Oct), B(Nov), C(Dec)
BDY
Body style		Each digit corresponds to a Make, Model and Body style
TRM
Trim		Indicates the style, color and material for the interior and seats
PNT
Exterior Paint		Indicates colors, 2-tone paint, etc. Each year has its own codes
A-Z
and 
1-0
Indicates engine and other options on the car