How are the basic Delphi types related to each other?

UInt8 = Byte
Int8 = ShortInt
UInt16 = Word
Int16 = SmallInt
UInt32 = LongWord
Int32 = LongInt
UInt64 = UInt64
Int64 = Int64

int = Integer
uint = Cardinal

NativeInt (generic, depends on CPU register size)
NativeUInt (generic, depends on CPU register size)

Cardinal and Integer are generic types. For 16 bit they were 16 byte large and for 32 bit they are 32 bit large. For 64 bit the Windows 64bit platform (LLP64) defines them as 32 bit. The new NativeInt and NativeUInt types are now the CPU register sized types.

Cardinal and Integer are aliases.

Cardinal ==> LongWord  (unsigned)
Integer  ==> LongInt   (signed)

The signed one-byte integer type is ShortInt. You can remember its size by the fact that it's not the same size as usual C implementations of the short type.

As for capitalization, capitalize the first letter. The documentation tends to leave the "int" part at the end lowercase, as in Longint, but I think it's more common to capitalize it. Don't write the types in all capitals unless you're using Platform SDK types and you want your code to show its C roots; otherwise I'd just write Word and DWord, Long and ULong, etc.)

Delphi 2009, perhaps earlier, already defines types like Int8 and UInt32. As for how to define Int and UInt, I'd say don't. The language you're using already defines Integer and Cardinal; don't introduce new type names when you don't have to. Keep the names you already have, and then everyone else will know what you're talking about. (Besides, Int is already a function in the System unit.)

Use Cardinal when you want an unsigned type and don't care about its size; use LongWord when the variable must be exactly four bytes. Likewise for Integer and LongInt. Use Cardinal when you want a four-byte unsigned type; use LongWord when you want a generic unsigned type and don't care about the size. Likewise for Integer and LongInt, nowadays. If you're writing 16-bit code, use LongInt when you need four bytes and use Integer when you don't care about the size; Cardinal and LongWord didn't exist in Delphi's and Turbo Pascal's 16-bit days.

The common wisdom for years had been that Integer and Cardinal would become 64-bit types on a 64-bit compiler, but that is apparently not the case. Instead, they will remain 32-bit types, just as their counterparts in Microsoft C++ do. Furthermore, there will be a new type, NativeInt, which will be a 64-bit type in a 64-bit compiler. The LongInt and LongWord types will become 64-bit types because they have always been the same size as the Pointer type, which was 32 bits even in 16-bit times.