A torsion angle (or dihedral0) is an angle around a bond. More generally, it is a relation between any four atoms that tells you something about their arrangement in space. Obviously this also applies to any four points, but this is not about the maths.
A protein's three-dimensional structure can be uniquely1 defined by the sequence of torsion angles along its chain. The backbone is defined by three angles - called φ(phi), ψ(psi) and ω(omega). The sidechains can be described by various χ (chi) angles, which I don't know a whole deal about, sorry.
The most often used angles are φ and ψ - which describe the angles around the bonds connecting the C-α to the N and C. Since the peptide bond is generally assumed to be 'flat' (defined as 180°) it can be ignored...mostly.2 Those two angles are generally plotted on a plane to produce a figure known as a Ramachandran plot3.
Calculation of a torsion angle involves maths that I don't understand, even if I have re-typed someone else's code that does the job. Think vectors, think dot product. Stuff like that. Even harder is translating a pair of torsion angles (say, -56°, -20°) to structure. Generally, students are taught about 'allowed' regions of the Ramachandran plot such as alpha helix (the example given in the last sentance) or beta strand (around -110, 100)4. So, it can help to think of a simpler system to aid visualization - namely butane. This is just four carbon atoms connected in a line, so all torsion angles of the molecule refer to rotation of the two methyl groups around the central bond. It also helps to draw (or imagine) a diagram common to chemistry textbooks, but maybe not to biochemistry, called a Newman projection. It is a projection down the central bond to show the rotation of the fourth atom (or group) relative to the first.
These diagrams show the steric effects that bulky groups can have on free rotation of the central bond. In proteins, this is mostly about the R group, which usually forces the φ angle to be negative. This is because all proteins (but not all peptides) are made only from left-handed amino acids. However, glycine is free to adopt almost any pair of φ, ψ angles - and it is this residue that is most often 'L' in proteins5.
Should you need to calculate these angles from scratch, you should know that φ for the ith residue is defined by t(Ci-1, Ni, CAi, Ci) and ψ is t(Ni, CAi, Ci, Ni+1). Also, all angles should be in the range [-180,180] - unlike some versions of Rasmol which have been broken and display angles like 514°!(grr).
0 I may have made an error here by not putting this writeup under dihedral, but I had to make a choice, and this is what I call these things!
1Okay, so this is only partially true. There are things like τ (tau) and so on that deal with deviations from tetrahedral angles for the Calpha. I'll ignore these.
2 Of course, variation does occur, and it can be important to note this - just not all the time.
3 Named after Professor Gopalasamudram Narayana Iyer Ramachandran who a) has a great name and b) sadly died in 2001. :(
4 Most pairs of angles are φ,ψ in that order.
5 In case you were wondering, L = positive φ while R = negative φ