There have been several posts with innovative ideas of what the phugoid is.
It might be useful to revisit the basics of the two pitch oscillatory modes .. both come from the mathematics involved in describing flight but can be understood adequately in a qualitative sense. The characteristics are aircraft design dependent and the motions will vary between Types.
Short Period Oscillation
The short period oscillation (known not very imaginatively as a SPO) is a bit like being on a boat in enclosed waters (ie a short fetch) where the wind waves are close together. If we were to look at the motion of the boat from the side we would see it going up and down comparatively quickly .. the period (time between successive crests) is reasonably short and the motion can be reasonably uncomfortable for the occupants.
For the aeroplane, the SPO is associated with a varying angle of attack and the two main concerns are
(a) the period, typically, is in the order of a second or so. As this is within the normal range of human response time, it is very easy for the pilot to excite the SPO (ie cause the magnitude of the oscillation to increase) by trying to control the initial motion (but getting into phase with it), leading to the phenomenon known as a pilot induced oscillation (PIO)
(b) associated with (a), loss of control or structural integrity are very real concerns
Design- and certification-wise, the SPO must be heavily damped for the aircraft to be acceptable.
Long Period Oscillation
The long period oscillation is a bit like being out in the ocean and looking at the effect of the ocean swell on our boat. From the side, the boat rises and falls with the swell over a comparatively long time.
The period, in this case, may be in the order of a minute or two. Generally, the period will be longer for larger aircraft, and shorter for the small machines.
The long period oscillation is characterised by an approximately constant angle of attack and is associated with a cyclical exchange of airspeed and altitude. If the aircraft is constrained to level flight, the motion will degrade into a cyclical speed oscillation.
Provided the long period oscillation is not divergent (ie the oscillatory motion doesn't get larger as cycles progress) the motion is only a bit of a nuisance for (and easily controlled by) the pilot.
Long period oscillation was the subject of research by Lanchester is the early 20th century and he coined the term "phugoid" for the motion (chaps of his era routinely were schooled in the classics and were much beloved of citing Greek and Latin roots in their naming of things ..)
Further reading
Any of the standard engineering undergrad Flight Mechanics texts will give a detailed mathematically modelled run down of both motions (although such knowledge is probably not overly necessary for the pilot) and I have no doubt that there are numerous useful descriptions on the Web.