If you are talking about PICBasic, then a rotation will push bits out the end and lose them... they are replaced with zero's being pushed back in...
Code:
' example 1...

a var byte

a=%11111111
a=a >> 1

' a now contains %01111111

' example 2...

a var byte

a=%11111111
a=a << 3

' a now contains %11111000
The result of an addition of two numbers will remain within the variable size you have decalred, and only the bottom eight bits (in a byte) or the bottom 16 bits (in a word) will remain.
Code:
' example 3...

a var byte
b var byte
c var byte

b=255
c=1

a=b+c

' Result in a = 0 (zero)

' example 4...

a var word
b var byte
c var byte

b=255
c=1

a=b+c

' Result in a = 256
If you need to check for an overflow, then you will need an additional line of code...
Code:
' example 5...

a var byte
b var byte
c var byte
overflow var byte

b=255
c=1

a=b+c
if a < b then overflow=1 else overflow=0