What is Planet? Why a Planet. Just a Planet.
The current semi-formal definition of a planet is: a body with a mass greater than or equal to Pluto and less than 13 times the mass of Jupiter that is orbiting a main sequence star or stellar remnant.
The controversy comes in two forms: one relates to the definitions of "planet" within the Solar System, and specifically whether Pluto is a planet or a "minor planet", merely the most massive of the family of Kuiper Belt Objects.
The second issue is for extra-solar planets; namely what is a planet outside the Solar System - that controversy is primarily related to issues of precedence, "who discovered planets first". Namely if you can define anything discovered before what you discovered to be NotPlanet, why then you discovered the first planet. Parenthetically, one should note that Nobel prizes are never awarded for being second. They go to first discoverers, method and technique developers, and "great body of work" and "amazing breakthrough" theorists.
So: in astronomy, objects are usually categorized phenomenologically - based on how they look, literally. This leads to embarrassing inconsistencies when we eventually understand the actual physics behind the appearance (eg Object X for which object class X is named, is in fact not an object of class X. Or, type Ib is a phenomenological class I, distinct from type Ia and in fact a part of class II...)
So, in the solar system, "planets" are resolved objects, that are round and orbit the Sun.
They have anomalous proper motion by virtue of orbiting the Sun, and they are finite sized, so resolved when observed through a telescope, and in words of a pioneering net.crackpot, they are not "space potatoes".
There is a minimum mass at which a rocky object undergoes elastic deformation under its own gravity to have a surface tracing a 2D contour of constant potential energy (isopotential). Minor planets are lumpy.
Conveniently, for historical reasons, Pluto is above this line and therefore a planet.
For obscure historical reasons, some people would like Pluto not to be a "real planet". Other people just strive for annoying consistency and want Pluto not to be a planet for such reasons.
One inconsistency is that this definition leaves the lower mass boundary for planets satisfyingly vague, it'd depend on the equation of state of the stuff making up the planet (since elasic deformation depends on composition) and low mass gaseous planets would have minimum mass that'd depend on temperature (cool).
So, for historical reasons a lot of people want to leave Pluto as a planet; for consistency reasons, others would like Pluto demoted and be the largest of a new class of minor planets.
Note that the most massive moons are already as massive as the least massive planets: in astronomy classes of objects overlap and have vague boundaries. Such is life.
Now, with extrasolar planets, life gets more complicated. For one, most of our observations are marginal and the data from which classes are formed have systematic (eg knowing M/sin(i) rather than absolute M) and random uncertainties.
Most people agree that a sensible boundary for the upper mass bound is "deuterium burning limit"; the mass at which objects sustain for a while thermonuclear fusion of deuterium in their core. For hydrostatic models of approximate solar composition, that boundary is at about 13 Jupiter masses. So, anything above that is a brown dwarf.
Bad news: if you look at core temperature profiles during formation, then depending on how and when the mass accreted onto the core, there may be temporary fusion of deuterium down to as low as ~ 3 Jupiter masses, if I understand the results correctly. People whose observations are biased towards the higher masses, especially with that annoying 1/sin(i) factor, tend not to like this "correction" to the definition so much.
Lower mass limit: well, we don't need to worry about that so much for extrasolar planets, yet. Only pulsar detections can probe that low, although other techniques will probe to lower masses surprisingly soon.
Some people feel that we should not define planets phenomenologically, the definition should be based on physical processes; one such being the formation mechanism.
Planet mass objects could form in three ways:
1) core accretion: whereby dust aggregated into dust-bunnies (in Spaaaace), into pebbles/snowflakes, into boulders, into planetesimals, into terrestrials, into super-Earths, and then at some (few) Earth masses there is runaway gas accretion into ice giant or jovian (or not, you could just stop at a lower mass for whateve reason).
2) disk instability: this is an old idea, recently resurrected for various interesting reasons. In this model, (some) giant planets form when a cold protoplanetary disk becomes gravitationally unstable and density maxima become self-bound and accrete gas to become jovian or superjovian planets. This may happen if the cooling time of such density structures is rapid enough, so it tends to happen in the outskirts of protoplanetary disks. The process is rapid; the resulting structure has either no rock or ice core, or a fairly modest one formed by segregation after formation. Process 2) could happen independently and concurrently with process 1), or they could both happen, with process 2) promoting process 1).
3) failed brown dwarfs: spheroidal instabilities in gas clouds lead to few jupiter mass (or less?) bound gas masses, which ought to grow to stellar size, but something interrupts (or even partially reverses) the accretion process and the star "fails" and is left as a low mass core with no fusion. This could happen in isolation, or in a dense star forming region, so the resulting core could end up bound to a star through dynamical processes.
Either way, I think it is already clear that the lowest mass brown dwarfs (ie H/He dominated spheres formed in isolation through spheroidal collapse of Jeans unstable gas) will overlap in mass with the most massive jovian planets formed in disks (and if you favourite core accretion scenario terminated formation at some safe mass like 3 or 4 jupiters, then we can always make "blue straggler planets" and collide two super-jovians formed in a disk to make a super-duper jovian of 5+ jupiter masses. Repeast as necessary...)
4) Something completely different. There are reasons to worry about anomalous and rare formation channels for planets to explain "exceptions", but not here and now. Yes, I know I said 3 at the beginning, that was deliberate.
So some people would like only objects formed by process 1) to be planets.
Problem is that it is very hard to observe a jovian mass planet indirectly and know how it formed (though there are some ideas on looking at the mass/radius relation, for cases where the radius is measurable, or composition, or even the effective damping constant to collective oscillations). Which is why astronomy like phenomenological definitions.
So the big outstanding issues for extra-solar planets are:
1) are "planets" only around main sequence stars or also around stellar remnants?
Well, people who observe main sequence stars would like planets to be only around stellar remnants. Then, conveniently, the first discovered planets (Wolszczan's pulsar planets) would retroactively not be planets and 51 Peg would be the first discovered planet.
However, this definition would require planets around white dwarfs not to be planets either. So when the Sun goes off the main sequence and completes its RGB and AGB evolution, the planets would no longer be planets even though half of them at least would still be there and be pretty much the same as before.
This is taking inconsistency to an unnecessary level (caveat: I am not a disinterested observer in this, so "bring it on", and I will test my newly discovered amazing jiu-jitsu bureaucratic powers...)
On a slightly different note: what about planets around brown dwarfs? They are easy to detect, but is a 5 jupiter mass object orbiting a 15 jupiter mass primary really a planet, or a low mass brown dwarf binary?
What about a 5 earth mass object orbiting a 15 jupiter mass primary?
2) are free floating objects with mass in the planet range actual "planets"? Well, no one, except the people who do IR observations of low mass free floating objects, wants to call them planets. They are sub-substellar objects.
Which is a bit annoying, since it means that you could make perfectly respectable planets around a main sequence star, of any appropriate mass, and in whichever way is approved, and then at an arbitary late time you could dynamically eject those objects into interstellar space, at which point they would abruptly not be planets anymore...
Consistency is, however, overrated.
Amusingly, there is a finite probability that we could observe a system whereby a planet has been formed, and is in the process of being ejected - in that it is on a dynamically unstable or even unbound trajectory but is still within the system...
Planet, or not planet? You decide.
(Odds are surprisingly good: timescale for the ejection is ~ 100 years or so, but a formally unstable system could be observed for 10,000 years or more. Young systems probably eject 1++ planets in the first 1-10 million years, so if we observe 10,000+ low mass protostars or pre/early main sequence stars with ages less than 10 Myrs, odds become pretty good we'd catch a system in the process of ejecting an excess planet).
Detecting interstellar planets is not too hard: they are IR bright when young; conceivably observable through mid-IR + parallax and/or proper motion at late times; and almost certain to be seen in microlensing searches.
There may well by between 10 billion and a trillion free floating planets (whose total summed mass is still negligible) in the Milky Way galaxy. Who knew.
Oh, and as David Stephenson noted, free floating planets can be habitable (liquid water on the surface) for interestingly long time (order Gyr). That'd be a twist.