Is it possible to use structures other than finite fields? I have some difficulties to understand why we are using finite fields in cryptography.

Why do we use field? Why not ring or group? Is that really necessary that the field is finite? Why not real field?
One does not always use finite fields in cryptography. RSA for instance is based on
residue class rings, discrete logartihm based cryptosystems work on suitable
additive/multiplicative groups but these are also finite rings/groups. 

Why one uses finite algebraic structures and not the field of the
reals? 


Elements of the field of the reals do not necessarily have a
finite representation and thus cannot be represented on computers
precisely (only approximations as floating point numbers of a certain precision). Consequently, field operations may only yield approximations of the resulting element. Working with such fields would introduce errors in the arithmetic operations, which we do not want in cryptography. Using finite fields allow to represent all elements precisely and also the output of any arithmetic operation in the field.
Furthermore, finite structures used in 
cryptography provide us with hard problems useful for constructing cryptographic protocols, such 
as the hardness of the discrete logarithm problem over suitable finite fields. Logarithms over the reals in contrast are easy to compute.