We must have a common language describing the motions that a vessel can make. It is easiest to first break down each motion individually by picturing a skewer penetrating a model of a little ship. The motion the vessel can make by rotating that skewer describes motion about a single axis. The axis through which a vessel may move are described as:
This is motion about the length axis, with the skewer penetrating our model stem to stern. Waves hitting the side of a vessel cause her to roll. This is generally the most uncomfortable of motions a vessel can make, excacerbating motion sickness, and depending on the stability profile of the vessel this motion can be very snappy and cause gear to come adrift out of closets or off of shelves. The nature of this motion can be lessened by quartering the seas, or taking the motion on either bow or on the stern while making a zig-zag pattern through the water.
If we leave the skewer in place, but now move it side-to-side, we have sway. A wave hitting the side of a ship may initially cause it to roll, but it may also push it back causing it to sway.
This motion is about the athwartship axis; picture the skewer entering one side of the vessel and exiting the opposite side, perpendicular to the ship’s fore-and-aft (or roll) axis. When our imaginary skewer is rotated, the bow goes up and the stern goes down, followed by the stern going up and the bow going down. The vessel generally has a great deal of momentum and weight behind her, which may be brought to a sudden stop by wave action. This motion is very punishing to the vessel. It may be mitigated by reducing speed or quartering.
If we pick the skewer up or lower it down, the vessel is said to be heaving. A vessel can heave at the dock through sudden wave action, for example, in addition to other motions she may experience.
With the skewer still in position for pitching or heaving, if we move it foreward and backward, the vessel is said to be surging. This is strictly the fore-and-aft motion of a vessel.
This motion is turning about the vertical axis, as now the skewer is entering the top of the vessel and coming out the bottom. When rotated, the vessel spins left or right. An example of yaw would be when being towed or when running before a following sea. Following waves may catch the stern and throw it to one side or the other. To lessen this effect, weight may be shifted aft to the stern, the vessel may be placed on a more quartering heading, or the vessel may be counter-steered–that is, you anticipate the direction of the effects of a coming wave and, with timing, negate its effects.