In botany, hybrid plants may be given a 'hybrid name' in place of the traditional species name. This is not a requirement, but may be used if an author feels that it will clarify matters.
Hybrid names are usually created by using the genus name of the two hybridized species, plus a unique species name marked by an leading X. For example, the common garden strawberry is Fragaria × ananassa, being a hybrid of two species in the genus Fragaria. The new name, in this case ananassa (the Tupi word for 'pineapple'), is chosen at the whim of the author, in this case presumably refering to aspects of both appearance and flavor.
In some cases the author may choose to use the prefix notho- + the appropriate rank instead of the X. This refers to a nothotaxa, an informal designation used to clarify. This prefix comes to us from the Greek nóthos, meaning 'bastard', although in this case usually translated as 'hybrid'. E.g. the Florentine Iris was originally given the formal name Iris florentina; when it was later discovered to be a hybrid form of Iris germanica, it was renamed Iris germanica nothovar. florentina. The 'notho-' system may have been chosen in part because the author wants to refer to a variety, not a species, hence the term "nothovar."; likewise, you may see "nothosect."
It is possible to hybridize organisms of different genera as well. In this case the author will have to come up with a name for the new genus as well as the new species. For example, the Shipova fruit is a hybrid between Sorbus aria and Pyrus communis. The new nothotaxa (in this case a nothogenus) is × Sorbopyrus, and the hybrid name is × Sorbopyrus auricularis. "Sorbopyrus" is known as a condensed formula, formed by combining the names of the two genera; in cases where this is not practical (e.g., when four or more types are combined), the name of the discoverer or developer is usually used.
Citrus fruit is famous for making many interesting and delicious hybrids, leading to compounded hybrid names. For example, the pomelo (Citrus maxima) has been crossbred to make many new species, including with the Mandarin orange (Citris reticulata) to form the orange (Citrus ×sinensis), and then again with the orange to create the grapefruit: the grapefruit is usually referred to as Citrus ×paradisi, but can also be specified as C. maxima × C. ×sinensis. Likewise, the orangelo can be specified as C. ×paradisi × C. ×sinensis (a grapefruit crossed with an orange).
It should be noted that within common hybrid species you may find dozens of cultivars; Fragaria × ananassa may be treated as a general synonym for strawberry, and there are untold variants that you can buy from any gardening catalog. These varieties do not each have a unique hybrid name.
This sort of naming is most often done for naturally occurring hybrids, but as you can see, it is not uncommon in commercial hybrids. Hybrid names are very much phylogenetic, rather than describing specific morphologies, as hybrids of a given set of species will share a hybrid name even if the hybridization produces a wide range of possible offspring, or if these offspring do not breed true over multiple generations.