Codes
Error Correcting Codes are used in almost all forms of digital storage and communication, from DVD players to satellite communication. The theory and practice of constructing good error correcting codes rely heavily on the abstract notions of finite fields and vector spaces (see e.g. Huffman[1]), and more recently on even deeper and more abstract algebraic concepts like function fields and algebraic curves (Stichtenoth[2]).
Expander Graphs
Expander graphs can be used to design sparse communication networks with very strong connectivity features. "Sparse" means that every node in the network is connected to just a few other nodes; "strong connectivity" means that in order to sever the connection between two substantial pieces of the network you'd have to cut lots of individual connections. Expander graphs are useful also in a variety of algorithms where one needs to simulate "randomness" in a deterministic way.
Constructing good expander graphs is hard. The best known constructions rely on very deep results in representation theory, see e.g. Lubotzky[3].
3D Modeling and Animation
Representing rotation in 3D space, particularly when one wants to interpolate between two rotation states as in 3D animation, is usually done using the algebraic concept of quaternions [4]. It's amusing to find chapters on "quaternions" in the user manuals of software such as Maya and 3d studio.
Block Designs
In a sense this is a generalization of the "Codes" example. Block designs are very symmetric finite set systems; a nice concrete example to keep in mind is the following little challenge: find a set of 7 triplets of numbers in the range 1..7 such that every pair of numbers in this range belongs to precisely one triplet (look up the Fano Plane).
Such symmetric designs are useful in a variety of real-life contexts such as the design of experiments (it's a good way to choose sample sets without biases) and - says Wikipedia[5] - software testing, though I'm not really familiar with how they're used for that.
Once again, constructing block designs relies heavily on tools from abstract algebra, specifically finite fields (especially projective and affine geometries based on finite fields) and finite groups.
[1] Huffman and Pless: Fundamentals of Error Correcting Codes.
[2] Stichtenoth: Algebraic Function Fields and Codes.
[3] Lubotzky: Discrete Groups, Expanding Graphs and Invariant Measures.