Aren't all semiconductors' atoms bond in a covalent bond? How is a "covalent semiconductor" different from other semiconductors?
No, most semiconductors are ionic.
I'm not sure I've seen the term "covalent semiconductor" before. Wikipedia has an article on covalent superconductors.
I think the main point would be that many if not most semiconductors are inorganic solid-state compounds. Consequently, most of these (e.g. III-V semiconductors are ionic solids, not really covalent ones. Consider GaAs or InP, which are typically classified as ionic.
That is the solid state electronic structure can be considered mainly composed of cations and anions (unlike organic chemistry where the electronic structure is considered mainly of neutral atoms).
(Incidentally, defining whether some of these compounds have purely ionic or purely covalent interactions is somewhat futile, since most bonding interactions are both somewhat ionic and covalent.)
Most semiconductors are inorganic solids but some polymers can be semiconducting and that is probably what the term "covalent semiconductor" is referring to.
The majority of semiconductor material are inorganic solids (as Geoff Hutchinson describes in his answer). The electronics industry is based on the use of silicon-based materials (often doped with small, controlled amounts of other elements to achieve the precise semiconducting properties required). Mixed compounds like GaAs or InP are also used. Arguing about the nature of the bonding in these compounds is is, as others have said, a little futile.
But there are a range of compounds that can be semiconductors that are more often thought of as "covalent". The most important of these are Organic Light Emitting Polymers (which, in devices, are called Organic Light Emitting Diodes or OLEDs). If the term "covalent semiconductor" means anything, this is the class of compounds it is referring to.
To simplify a little, the semiconducting property is created by delocalisation of electrons along the pi-system in the molecular chain as opposed to by the overall crystal structure of an inorganic solid.
There is a good Wikipedia article on the topic under the title "Organic Semiconductor". These are now becoming significant in display technology.
Pure silicon (Si) and pure germanium (Ge) are examples of covalent semiconductors. To explain, first take a look at the difference between ionic and covalent bonding.
In ionic bondig, the electron is given by the least electronegative atom and taken by the most electronegative atom. If the difference in this electronegativity of both atoms is rather large (so atom A wants to give badly, atom B wants to take badly), the bond is regarded as strictly ionic.
On the other hand, if there exists a bond between two atoms which have a similar affinity for an electron, the bond is regarded as covalent. This must be the case in pure silicon, as there are no other kinds of atoms present, so there are no different affinities. Silicon has 4 valence electrons and forms a crystal structure, sharing it's 4 valence electrons with 4 neighbors, thus all filling it's orbitals to the maximum of 8 electrons.
Silicon is an intrinsic semiconductor, meaning that without any modifications, some electrons may reach the conduction band. Adding dopants to a crystal structure such as this, can change these semiconducting capabilities. The difference with oxide semiconductors is that in covalent semiconductors, the effectiveness of dopants at a given temperature is solely determined by their ionization energies. In oxide semiconductors, it also depends on the extent of ionization of the structure. After all, the presence of the dopant in an ionic compound (oxide semiconductor) can be partly charge-compensated by other ionic defects. This is explained in chapter 2 of 'Physical Ceramics'.
A covalent semiconductor is one whose crystal structure contains covalent bonds. A covalent bond is one in which electrons are equally shared between the bonded atoms. Only silicon and germanium are purely covalent because all bonds are identical and thus, by symmetry, electrons are shared exactly equally. Compound semiconductors like gallium arsenide, GaAs, are partially covalent, there is some imbalance to how the electrons are shared in bonds. They spend more time with one of the constituent atoms than with the other one.
On the other end of the scale are ionic bonds and ionic solids where one atom in the compound rips the electron from the other atom. The electron then essentially becomes part of the negatively charged ion. NaCl is such a solid. I can’t think of any ionic semiconductors.
In Introduction to Solid State Physics, Kittel has a table of the “Fractional ionic character” of bonds in crystals. Silicon is 0 ionic (meaning fully covalent). GaAs is 0.31 ionic and NaCl is 0.94 ionic. All of the III-V semiconductors are in the range of 0 to 0.4. Some semiconductors like CdS go as high as 0.7.