If you're using alerts in Apple Calendar, you will see a little box pop up on the upper right hand top corner of your screen. If you don't have time to attend to it right away, click Snooze. If you have already done whatever it is that the alert has for, then click Close. If you haven't done that thing yet, don't click Close as it's the same as turning off an alarm, it makes it go away permanently.
I've been teaching myself to use these functions in iCal by doing little silly stuff, like notifying myself to change my house filter. I find that it's the best way for me to learn, by doing. As I get more comfortable with it, I will lean more heavily on it. If it's new to you, then double-check things by going to your calendar, or your Notifications on your desktop or iPhone. On your desktop, you click on the little icon that looks like a list at the top far right of the menu, on your phone you get to Notifications by sliding down on your screen and then selecting Notifications.
You can also just leave the alert up, but you shouldn't do that, as you will very quickly learn to ignore it. So instead either Snooze or deal with it. If it's your house filter, it can wait, if it's someone waiting to hear a response to a meeting, take a look at your calendar and accept or decline right away. If you're organized, this shouldn't take you more than two minutes. If it's taking you longer than that, you need to reevaluate how have your workflow set up. I keep my Calendar open and all I need to do is to click on it in the dock and it's instantly visible, there's no need to wait for the software to launch.