A rogue planet (or a rogue anything, for that matter: a celestial body other than a star) is something that's drifting through space without being attached by gravity to any star. They're just out there in the big black void. Is there any evidence that they actually exist? Or are they purely hypothetical?
Rogue planets have been discovered by infra-red imaging because planets are hot when they form. Here's a list of a few.
There's 2 types of Rogue planets. One is failed stars. condensing pockets of gas and dust that form similar to how our solar-system formed, but that are too small to form stars. Source.
The 2nd type is planets that escape from a star's orbit. This can happen by the star ejecting material and losing mass which causes the planets to expand in more distant orbits, some eventually escaping, or by gravitational assist either planet on planet or by two stars passing quite close to each other. It's statistically impossible for Rogue Planets not to, from time to time, get ejected from solar systems, so they have to exist at least in reasonably large numbers throughout the galaxy, though I don't think it's well known how common there are.
Our solar-system might have ejected a planet - see here. In general, larger planets can eject smaller ones but mostly not the other way around, but two stars that pass too close to each other can eject any planets, mostly ones with more distant orbits. A star that loses a lot of it's mass or a star that goes nova can push planets out of it's orbit too.
Giant planets when first formed are big and hot. They radiate their own light, mostly in the infrared. So young isolated planets can be seen directly.
There have been various claims in the literature that objects as small as a few Jupiter masses have been identified in young star forming regions. See various papers by the IAC brown dwarf research group
Another object that is part of the beta Pic moving group, recently discovered by Liu et al. (2013), has an estimated mass of about 8 Jupiter masses (Biller et al. 2015).
These claims are open to criticism - sometimes it is hard to tell whether a faint object really belongs to the star forming region observed, rather than being an unassociated background object. The claimed masses also depend heavily on models for the luminosity-mass relation as a function of age, and the ages of these objects are not easily constrained. The likelihood is that at least some of these objects are below 10 Jupiter masses and would rank as planets by some definitions; though none of the individual objects could be said to be proven beyond any doubt.
Nevertheless it would not be surprising if, in the maelstrom of the formation of a cluster of stars, some planetary systems were stripped from their parent stars by close encounters with other objects and indeed numerical simulations of planetary systems in dense star clusters show that this process occurs (e.g. Davies 2011).
The chances of seeing older, isolated, planetary mass objects are slim, but microlensing appears to be the only technique presently available. The microlensing signature of a free-floating planet is of course unrepeatable so a discovered planet could not be followed up in any way. However, surveys of microlensing events could be a way of saying something statistically about how common such objects are. See for example http://astrobites.org/2011/05/24/free-floating-planets-might-outnumber-stars/
It is also worth noting that the whether these things really are "planets" at all is disputed. They could either be genuine planets, formed in the same way that is hypothesised for most giant planets - that is by accretion onto a rocky core that formed around a star. They could then have been displaced from their parent star by dynamical interactions with other bodies in their system or with a third body. As I said above, N-body simulations do predict that this will happen (e.g. Liu et al. 2013).
On the other hand they could represent the very lowest mass gas fragments that are able to form during the collapse and fragmentation of a molecular cloud and that for some reason were unable to accrete further gas (i.e. they are really more like low-mass brown dwarfs). This so-called "fragmentation limit" is of order 10 Jupiter masses, but if it were a little lower it might explain the free-floating "planets" that have been seen so far.