"Congratulation" vs. "congratulations"

Solution 1:

Congratulations is simply the plural form of congratulation. See these examples from the Merriam-Webster dictionary:

1. Let me offer you my congratulations for being elected.
2. Please send her my congratulations.
3. I sent her a letter of congratulations.

The plural form illustrated by the examples above is much more used than the singular form:

2523 matches for congratulations vs. 56 matches for congratulation in the COCA.

It should be noted that the less common singular form can be found in sentences where the plural form would not apply. See these examples extracted from the COCA:

1. ...notes of congratulation and consolation.
2. The Artistic Podiatrist extended a hand in congratulation.
3. I made conventional sound of congratulation.

Solution 2:

Although Bruno's examples are quite correct, I have very rarely seen the singular "congratulation" used in practical English. Any time you refer to offering someone congratulations you use the plural and I think you could easily pass yourself off as a native English speaker without ever using the singular.