Subhorizon vs superhorizon In this review (page 18 top) it is argued that superhorizon modes are defined as $k_{\rm phys}<H(t)$, while subhorizon modes are defined as $k_{\rm phys}>H(t)$.
What is the physical understanding of this distinction?
One also speaks sometimes about that modes can enter or exit the horizon. Is this related?
 A: A mode $k$ is associated with a wavelength $\lambda\propto 1/k$ so when a mode is $k < H$ it is said to be superhorizon  meaning that the wavelength is larger than the Hubble radius, because Hubble radius is proportional to $1/H$.
Be careful, the Hubble radius is no horizon but it's common to say that if something is larger than the Hubble radius then it's super horizon.
About the saying "entering the horizon". This is an unfortunate terminology but it's a standard one.
Think of a mode larger than the Hubble radius, for example think of a perturbation in the density field with a wavelength larger than the hubble radius. Now, the Hubble radius gets larger and larger with time, so at a certain point the Hubble radius will be as large as the wavelength associated with that perturbation, this is said to be the moment of the horizon entering even if it's actually the Hubble radius getting larger and "eating" larger and larger scales.
Think of it in this way, far far back in the past every mode is superhorizon, then the Hubble radius grows and it gets larger than some modes, it is then said that these modes entered the horizon. Some mode are still out of the Hubble radius, but they will enter the horizon at a later time.
This distinction is important because superhorizon and sub-horizon modes evolve differently, and another crucial thing is that the evolution of the modes after they enter the horizon strongly depends on when they enter the horizon during the history of the Universe.
I hope I have been clear enough, if not, please ask me in the comments and I'll try to explain myself better
