Difference between backbonding and hyperconjugation and conjugation
OK, let's step through these terms and let's add &quot;cross-conjugation&quot; and &quot;homoconjugation&quot; for good measure to round out the series.
Backbonding:
This term is most often used to describe a type of bonding that occurs in inorganic compounds.  Metals bonding with carbon monoxide provide good examples of backbonding. The carbon atom in carbon monoxide has a lone pair of electrons that can be used to form a sigma bond with a metal.  Because carbon monoxide has low-lying $\pi^{\ast}$ orbitals, it can accept electrons back from the metal and further strengthen the bonding between the metal and the carbon monoxide ligand.  This process of &quot;accepting electrons back from the metal&quot; is termed backbonding. Here is a good picture of the $\ce{\sigma}$ and $\mathrm{d}-\pi^{\ast}$ (backbonding) bonding interactions.

It happens most often when 1) the metal is electron rich, 2) the ligand has low-lying $\pi^{\ast}$ orbitals and 3) the metal has available a $\mathrm{d}$ orbital with the proper symmetry to overlap constructively with the carbon monoxide $\pi^{\ast}$ orbital.  Here is a link to a great, one page article on backbonding if you'd like to read more.
Hyperconjugation:
This term is most commonly used in discussions about resonance.  Hyperconjugation is a stabilizing interaction that results from the interaction of the electrons in a $\sigma$-bond (usually $\ce{C-H}$ or $\ce{C-C}$) with an adjacent (empty or filled) $\mathrm{p}$-orbital. This interaction creates an extended molecular orbital that increases the stability of the system.  Most typically it involves the breaking of a carbon-hydrogen bond and then using what were the $\ce{C-H}$ bonding electrons to stabilize an adjacent charge. In resonance structures, we can't move nuclei, but we can move electrons. For example, below is a diagram showing resonance structures that involve hyperconjugation that serve to stabilize the n-propyl carbocation. Hyperconjugation has been used as part of the explanation for the carbocation order of stability being tertiary &gt; secondary &gt; primary.

Recently it has been suggested that hyperconjugation, rather than steric effects, is what stabilizes the staggered conformation of ethane (very interesting!).    Here is a link to a fairly extensive, but very well written article on hyperconjugation, it covers the effects of hyperconjugation on resonance structures and conformational stability.
Conjugation:
This term is used to describe a system of continuous, overlapping p-orbitals where electron delocalization can occur.  Examples of conjugated systems include the 1,3-butadiene, acrolein, benzene, beta-carotene, etc.


Cross-conjugation:
Sometimes you'll hear the term &quot;cross-conjugation&quot; used.  In this case, the continuity of the overlap between the p-orbitals does not follow a simple linear pattern as seen in the examples of conjugation pictured above.  Benzophenone and 3-methylene-1,4-pentadiene are examples of molecules with cross-conjugation.


Homoconjugation:
Sometimes $\mathrm{p}$-orbitals that are not directly conjugated (e.g., that are not directly connected, there is a &quot;spacer&quot; in between them) can still interact or overlap in a &quot;through space&quot; fashion.  The solvolysis of the exo- and endo-norbornenyl chlorides shown below provides a striking example. The exo-isomer generates the corresponding carbocation thousands of times faster than its endo-isomer. This difference is attributed to homoconjugation. The $\mathrm{p}$-orbitals on the remote norbornenyl double bond can interact through-space with the developing $\mathrm{p}$-orbital that is being formed on the carbon where the exo-chloride is leaving from, dramatically stabilizing the formation of the resultant carbocation. This interaction is geometrically impossible for the endo-isomer.  Again note, there is no direct conjugation between the remote double bond and the carbon atom where the carbocation will form, it is entirely a through space interaction that generates a pseudo-allyl system.