A wave packet is a localized wavelike perturbation, which appears frequently in quantum descriptions of particles. For instance, incident x-rays during scattering experiments can be thought of as wave-packets.
Note that these two descriptions are Fourier transforms of one another. Thus, there is an inherent tradeoff between the 'spread' of a wave-packet in position-space vs. momentum-space.
In the context of quantum mechanics, particles can be thought of as wave-packets. That is, quantum particles are neither ideal point-like particles, nor ideal plane-waves. They are instead intermediate objects, which are certainly wave-like (e.g. can undergo interference), but also somewhat localized. The classical concepts of 'particle' (perfectly localized; mathematically a delta-function), and 'wave' (oscillation with a single well-defined wavelength, spread infinitely over all space) can be viewed as limited cases of the general wave-packet. These limiting cases are only conceptual: in reality neither can exist.
In quantum mechanics, the wave-packet necessarily has both real and imaginary components.
Propagation and Interaction
Wave-packets can propagate, frequently with dispersion (which induces spatial spread of the envelope). Wave-packets can interact with barriers (defined by, e.g., some potential function), and interfere with themselves.