Q:

a banana and an apple is correct

I have to make slide where I have to write database and application word, so I want to know which sentence is correct? 

There is a database and an application. 

or

There is a database and application. 

Actually I want to know and conjunction rule in english language when consonant and vowel word is together in a sentence. 
Ps.  I do not want to change the order of the word in this particular sentence. 

A:

There is a database and an application. 

This sentence is correct.  This is explicitly separating or enumerating parts. 

There is a database and application. 

This sentence is also correct.  This is implicitly suggesting that these two parts are parts of one whole. 

There is a database and a application. 

This sentence is incorrect. " A" application is not grammatically correct; for nouns beginning with a vowel, use the indefinite article "an" instead. 

There is an database and an application. 

This sentence is incorrect. " An" database is not grammatically correct; for nouns beginning with a consonant, use the indefinite article "a" instead. 
An aside:

There is a databases and an applications. 

This sentence is incorrect.  For plurals, you do not use indefinite articles.  In addition, the verb is must be pluralized as well to are. 

There are databases and applications. 

This sentence is correct.