The original text:
We seem to forget that we all have some rights over the government. The government has come into being primarily to serve the needs of the citizen, which [sic] he as an individual or as a member of a small community cannot take care. It is important (that) we have an accountable government.
I really don't understand why "that" is correct the first sentence and not second?
Let's get this straight: that is optional in both sentences (as said in comments under your question).
There are several cases that that could be omitted. (Some people call this "omission", some call this "ellipsis".) Your text is an example of two of them. The first one is an omission of that after a reporting verb. The second is an omission of that after an adjective.
that: omission after reporting verbs
This is common after verbs such as say, tell, learn, discover, find, know, feel, think, etc.
He said (that) he felt better.
She told us (that) she was sorry and left the room.
We learned (that) the locals believed photographs would steal their souls.
We discovered (that) they had stolen the bag.
He was surprised to find (that) they had arrived at the bank.
Your first that falls into this category of usage.
We seem to forget (that) we all have some rights over the government.
Also note that you cannot drop that after some reporting verbs such as shout and reply (though some dictionaries seem to suggest that that after reply is optional, e.g. 'reply (that)'
in this dictionary).
She shouted that she was leaving now. (NOT
She shouted she ...)
that: omission after adjectives
In the pattern be + adjective + that-clause, particularly when the adjective is related to feelings or opinions, that can be (and usually is) omitted. Adjectives commonly used like this are glad, sure, sorry, afraid, surprised, pleased, important, necessary, essential, etc.
I'm glad (that) she made it.
I'm sure (that) he can handle that.
We were afraid (that) none of us would survive.
We were surprised (that) they came.
It's important (that) you understand.
It's necessary (that) we talk with them now.
Your second that falls into this category of usage.
It is important (that) we have an accountable government.
Caution: that is required if your subject is a clause
Note that if you use a clause as the subject in a sentence, you cannot omit that, e.g., That we'd survived the trip was something of a miracle. This sentence has two clauses we'd survived the trip and that was something of a miracle. When we write it as a single sentence, with the first clause as the subject of the sentence, we must use that, i.e., That we'd survived the trip was something of a miracle. (This type of that-clause is also known as "content clause", as Araucaria pointed out in a comment below.)
Bonus: 'that': not too many, not too few
Because that is optional, it's a fair question to ask: when it's possible, should we or should we not omit it? In order words, how should we decide when to omit that (when it's possible)?
Unfortunately, there is no quick and dirty rule. It's easy to imagine that someone would omit that whenever they can, whereas someone else would keep that whenever possible, and most people would be somewhere between the two extremes. The bottom line is you have to use your own discretion.
However, there is a good rule of thumb: do not omit that if ambiguity results.
These examples could help explain why:
He said that after the show was over the committee would award the winner the medal.
= The committee would award the winner after the show.
He said after the show was over that the committee would award the winner the medal.
= He said that after the show.
He said after the show was over the committee would award the winner the medal.
Ambiguity alert! Did he say it after the show? Or would the committee award the winner after the show?
Also, keep in mind that too many (of thats) is as bad as too few.
If you believe that keeping that is always safer, you could fall into this kind of trap:
He said that after the show was over and it had been officially announced that the committee would award the winner the medal.
(This is arguably ungrammatical, even, if the writer tries to elaborate a bit more of the same example above.
Can you see the ambiguity? ;-)