(To understand this concept I recommend you first refer to the following nodes 1, 2, 3, 4 and 5 and make sure you know what an interpretation function is. If there are any other nodes with good explanations of the language model relationship please /msg me.)
A relational structure is a kind of mathematical structure or model that only has relations (sometimes also referred to as a relational model, just be careful when talking to a modal logician since they mean something else by a model). I.e. it forms the semantics for a language that has no functional symbols (such a language is sometimes called a relational language). Therefore a relational structure would be some kind of set S with an interpretation function that interprets the relational symbols from the language as subsets of S.
Kripke models are a much used example of this. As an example we will consider a simple Kripke model viewed from an algebraic logician's perspective. This Kripke model will have two objects W and R where W is the set of worlds of the model and R is a binary relational symbol. We also need an interpretation function I that goes from the syntax of this model (which has variables and R) to subsets of W. In practice I will intrepret R in such a way as to show which worlds are reachable from other worlds. So let's say that a and b are worlds (i.e. a and b are elements of W) then aRb would mean that you can reach world b from world a. (Formally we would write I(R(a,b)) but that gets hard to read...)
Related mathematical subjects where these kind of structures get used include model theory, modal logic and formal methods.