AdWords has a tool built in called Ad Preview and Diagnosis. This tool allows you to preview your ads and determine whether your ads are currently being displayed in a specific area for specific keywords.
It is recommended you use this tool rather than performing an actual search on Google.com. Searching on Google.com will add to the impression count for your ads and keywords (which will lower your click through rate). The Ad Preview and Diagnosis tool does not have this affect.
The tool will also show you which of your keywords is triggering the ad for a given search query (this can sometimes be surprising). And if your ads are not showing, this tool will why so you can work to address the reason.
To get to this tool, log into AdWords and click Tools on the top menu. From the dropdown menu, select Ad Preview and Diagnosis. You can also pull up the tool by clicking here: http://www.google.com/adpreview.
On the next screen, start by entering your search term in the top box. Under Additional options enter a location (select a location that is as specific as possible). You can also select which type of device you want to preview (desktop, mobile, or tablet). Click the Preview button and then scroll down to see the results.