Try doing vs. try to do

Try + to + infinitive (try to do)

I make a conscious effort, be it physical:

•    She tried to move the cupboard on her own but it was too heavy so she decided to wait for her husband to come home.
•    I tried to hear what they were saying in the other room but I only managed to catch a few phrases. 
•    I tried to open the door but it was locked from the inside.

or mental:

•    He tried to understand her but she just cried and cried and never said what the problem was.
•    My mother tried to teach me some German when I was little.

In other words, I try to do something that is difficult.

Try + gerund (try doing)

Attempt. I experiment to see if it works. Normally used when I try to solve a problem:

•    It was really hot last night. I tried opening the window but it didn't help.
•    If she doesn't return your calls, why don't you try sending a letter and some flowers?
•    I tried adding a little salt, like the cookbook suggested, and it really made the dish taste better!

That is, I do something in an attempt to reach a certain effect. The focus is not on effort, but on attempting one approach or other.

Comparison:

He tried to read the newspaper but he couldn't because he had forgotten his glasses. (Reading without glasses was really difficult.)
He tried reading the newspaper to take his mind off his problems. (He wanted to find a distraction and the newspaper was a possible one.)