Solitons are just nonlinear waves. They appear in almost any nonlinear system, similar to usual (linear) waves that characterize excitations in different systems (deformation waves, acoustic waves, electromagnetic waves). A distinguishing feature of a soliton is that it is localized in space. Usually, a soliton has a bell-shaped form (sometimes, this type is called "dynamical soliton"), or a shock-wave or kink form (called as "topological soliton"). Another feature of the soliton is that it behaves like a particle, when interacting with another soliton or some obstacle (potential). In early days (in 60-80-s of 20th century), a term "soliton" was referred to excitations in integrable systems (such as the sine-Gordon model, the KdV equation, the nonlinear Schroedinger equation etc) only. But today, this term is applied to almost any particle-like excitation in different nonlinear systems.
For beginner, I would recommend a book by M. Remoissenet, "Waves called solitons", which is a good introduction to the topic. Also, there is relatively old, but still good and short book by P.Bhatnagar, "Nonlinear Waves in One-Dimensional Dispersive Systems".