how are two groups of an organism contain two different genus names but have the same family and what is this?

Answer 1

In scientific classification, family falls before genus. So it is quite possible for two groups of organisms to have same family, but two generic names*.

Look at the two classifications of Tiger and Cat.

Tiger:
Kingdom: Animalia

Phylum: Chordata

Class: Mammalia

Order: Carnivora

Family: Felidae

Genus: Panthera

Species: P. tigris


Cat:
Kingdom: Animalia

Phylum: Chordata

Class: Mammalia

Order: Carnivora

Family: Felidae

Genus: Felis

Species: F. catus


So you see, cat and tiger are closely related because they have the same family.(felidae). But they fall into two different genera*.

Now look at the classification of leopard, and compare it with tiger's and cat's.

Leopard:
Kingdom: Animalia

Phylum: Chordata

Class: Mammalia

Order: Carnivora

Family: Felidae

Genus: Panthera

Species: P. pardus

You see, leopard also falls into family Felidae so it is also closely related to cat and tiger. But its genus is different than that of cat's, and as same as of tiger's. So it is more closely related to tiger than cat.

Got it? The lower their common categorization in the classification system, more closely related they are.

*A little rule to add into your knowledge:
genus is a singular noun. The plural is genera. And the adjective is generic.

Answer 2

The classification system has these seven main levels: kingdom, phylum, class, order, family, genus, species. Kingdom is the largest grouping, and each kingdom is divided into phyla. Looking down the list, a family is divided into several genera. A genus is divided into different species.

Answer 3

Family is a grouping above genus, so it is quite common.