The act or fact of entering a train.
--OED
'To entrain' does indeed mean to get on a train, but more generally it means 'to drag along'. It comes to us from the French traîner, meaning to 'to drag' (this is the same root from which we get our word 'train'). In many technical jargons it is commonly used to mean just this, but over time the word has widened its meaning, so the word 'entrainment' may seem quiet confusing to someone who is not familiar with the jargon. While I certainly can't cover every usage of the word, here are some of the most common ones.
Rivers and glaciers entrain sediment as they flow along, picking up whatever they can carry and adding it to the flow. Glaciers and ice sheets can entrain giant boulders, as the freezing ice forms around existing landforms and then starts to flow downhill, ripping chunks out of the landscape. Rivers tend to move smaller particles, but they do it much faster and the sand and dirt that a river carries along with it can drastically change a landscape. Entrainment is a major factor in erosion and weathering.
Air entrainment refers to air bubbles being introduced into a liquid, such as molten metals or especially concrete. In this case, entrainment is a measure of how 'bubbly' the finished, hardened product is. Concrete may be strengthened by the entrainment of air bubbles, so a surfactant may be added to the mix to encourage air bubbles to form. Metal-casters, on the other hand, want to avoid air bubbles, as they will weaken the finished product. When talking about air bubbles being picked up in water and other substances that are liquid at room temperature we usually call it aeration, not entrainment.
But liquids aren't the only entrainers. Gases can also entrain each other. In fluid dynamics and related fields (including meteorology), entrainment refers to a turbulent fluid flow affecting a second fluid flow. In other words, when a turbulent flow is adjacent to a less turbulent flow, the turbulent flow will advance into the less turbulent flow, entraining it and making it more turbulent. Note that this is not a 'jet' of fluid hitting a stagnant 'pool' of fluid, but simply one flow passing adjacent to another flow (or stagnant layer). In meteorology this accounts for the mixing of otherwise discreet airflows, which can often mix humid and dry air masses.
In some fields of physics entrainment refers to the effect in which two interacting oscillating systems will synchronize themselves. Two pendulums connected through a spring might synchronize to swing in antiphase to one another, or two metronomes might match phase (in this case 'spring' is used in the strict sense, meaning any system that relates force to displacement, i.e. that tries to move back to its resting position). Electromagnetic waves also oscillate, and will align in the same way as mechanical systems. It's worth noting that in acoustics this synchronizing effect is called sympathetic resonance, not entrainment.
Neuroscientists may refer to brainwave entrainment. This is closely related to entrainment as it occurs in physics; essentially brain waves can be manipulated through periodic sensory input. When listening to periodic beats of the right frequencies EEG readings will start to change so that they are more in sync with the auditory stimuli. This can be used to increase alpha waves, which reduce stress. However, it is not clear that simply listening to music will have this sort of effect. Most studies have focused on binaural beats (different beats in each ear), and have concluded that the cause of entrainment is the different resonances in each of the hemispheres of the brain; when these signals interact the brain perceives it as a pulse equal to the difference to the two beats. It is this interference pattern that the brain actually aligns to. While this has been scientifically tested and confirmed, there are a lot of 'new age' programs out there that try to apply this in ways that have not not been scientifically confirmed, so beware. Photosensitive seizures are also set off by a process that could be called entrainment, but it is not generally referred to as such by those studying epilepsy; the term 'photoconvulsive response' is used instead.
In medicine in general entrainment refers to biological cycles changing to match rhythmic stimuli, both internal and external. For example, within your body locomotor, respiratory, and cardiac rhythms tend to synchronize (an example if internal stimuli). It is probably most common to hear it in reference to electrical stimulation of the heart to stop or prevent arrhythmia (this would be an example of an external stimulus).