First review how to open/clean your XBox 360 Controller, here
Symptoms: My stick is flacid, lol. There is no resistance or there is too much. Sometimes I will hit 'bumps' when moving my joystick. My joysticks alternate button doesn't push down or click properly. My joystick moves very slowly or is unresponsive.
Fix for All XBox 360 Joystick Issues.
After you have taken the controller apart the first thing you want to do as always is clean the joysticks base thoroughly. Including the "press down" button on the joystick, which is located on the right side of the joystick's base, also the inside should be cleaned with a Q-Tip and if available, canned air to blow out any debris. Once it is cleaned thoroughly, check to see if the problem is still there. Double check for any debris by rotating the stick 360 degrees slowly. For any obstructions that may need to be removed.
If the problem is to due with the joysticks movement and cleaning it thoroughly didn't fix it then you might need to replace the joystick, you can find that guide here. All it requires is an original XBox controller that you don't mind taking apart. Many pro's prefer using the older style sticks due to their extreme sensitivity compared to the 360's model.
If the problem is a stick/malfunctioning joystick "button" look closely at what is happening when you press down the button on the joystick, check for any sticky gunk surrounding the button on the right side. If any gunk is there that is more then likely causing the issue and a simple cleaning will fix it. Otherwise it could be a bad spring, check to see if pressing the button directly (instead of the stick itself) makes it click, if not you will need to replace the entire unit and that guide can be found here.
If this didn't solve your issue and you have any questions about the steps please leave a comment with the steps you have taken and I will try to assist you further.