How does a country get influcened by another country?

A country can get influenced by another country if one country has somethig that the other country wants or if one country is more powerful it can then influence the second country to do what the first country wants it to do.

It depends what aspects you look at it from. There are many reasons. Here are some off the top of my head.
Religion:
In the Middle Ages, Europe was influenced by Italy (Rome) as it was where the Pope was situated, and if you upset the Pope then you would go to hell (obviously, in the 21st century we know that was a load of twaddle, but back then it was not).
Money:
If a country is a good source of money, or has something that is worth a lot of money, then you will treat them well. I.e. if you meet a country full of oil, and the country has many weapons but does not want (all or some of) the oil, you would not attack them as you would get no oil, but you would treat them well to try and flatter them to make them give you the oil.
Military might:
If the country is strong, you would not want to get on the wrong side off them, and would probably want them as allies. Alternatively, if you see them as too much of a threat, you may be compelled to attack them.
The other countries allies:
If the other country has a lot of allies, you wouldn't want to upset them, and would also want them to recommend them as potential candidates / trustworthy traders to their allies.
Culture:
Another countries culture and beliefs can influence relations between them.
Defence:
Another country may be an enemy with one of your enemies, and so may be a potential wartime ally. Or, may be a permanent ally, which would be very beneficial.
Mutual advantages:
A case in which one country has the solution, but the other has the resources to make it happen, may result in an alliance.
There are probably more, but those are the main ones.

Here's a list of ways that one country might influence another:
(1) Geography -- A country's place on earth helps determine the other countries that influence it. This is why Japan and South Korea are very concerned about North Korea; whereas Kenya and Peru are not as concerned.
(2) Culture -- Samuel Huntington's "Clash of Civilizations" (see source) groups countries according to cultural blocs. American support for Israel, as well as the unity of Arab opposition to Israel, are (at least partially) explained by cultural factors.
(3) History -- The United States and Great Britain share deep historical ties. Russia, remembering the lost empire of the former Soviet Union, has exercised its power over the former Soviet republics.
(4) Economics -- Oil plays a role in American policy in the Middle East. China is gobbling up resources in Africa.
(5) Military power -- America is currently using this type of influence in Iraq and Afghanistan. Russia invaded Georgia. Iran and North Korea have nuclear weapons programs.
I'll stop here for now. My answer is by no means complete. Feel free to ask follow-up questions if you want. Your question is very well worthy of further consideration.