Normally there's no benefit to using different sub-types of uints (uint8, uint16, uint32, etc.) because solidity reserves 256 bits of storage regardless of the uint size. For example, using uint8 instead of uint (uint256) won't save you any gas.
But there's an exception to this: inside structs.
If you have multiple uints inside a struct, using a smaller-sized uint when possible will allow Solidity to pack these variables together to take up less storage.
Example
struct NormalStruct {
uint a;
uint b;
uint c;
}
struct MiniMe {
uint32 a;
uint32 b;
uint c;
}
// `mini` will cost less gas than `normal` because of struct packing
NormalStruct normal = NormalStruct(10, 20, 30);
MiniMe mini = MiniMe(10, 20, 30);For this reason, inside a struct you'll want to use the smallest integer sub-types that will suffice for the struct.
Clustering
You'll also want to cluster identical data types together (i.e. put them next to each other in the struct) so that Solidity can minimize the required storage space. For example, a struct with fields uint c; uint32 a; uint32 b; will cost less gas than a struct with fields uint32 a; uint c; uint32 b; because the uint32 fields are clustered together.