1) First, you'll need to create a filter so that Gmail will automatically assign a special label to your alert-worthy messages. Sign into the regular Gmail Web interface from any computer. Click the small down arrow at the right of the search box, then fill in whatever info describes the type of email you want to target for your alert -- a specific sender's name or email address, a specific keyword in the subject, or whatever the case might be.

2) Click the link at the bottom of the search box that says "Create filter with this search." Check the box that says "Apply the label" and then create a new label. You might want to call it something like "ATTN" or "Notify."

3) Be sure to click the blue box that says "Create filter" to save the rule when you're done.

4) Now pick up your Android device and head into the Gmail app. Tap the overflow menu icon (the three vertical dots) at the top-right of the screen, select Settings, and then select your Gmail account from the list that appears.

5) Scroll down and select the "Manage labels" option. Find your new label in that list -- "ATTN," "Notify," or whatever you ended up calling it -- and tap on it.

6) Tap "Sync messages" and change the setting to "Sync: Last 30 days." Then check the box for "Label notifications" and tap the "Sound" option to pick what sound will play when one of your important messages comes in. You can also select whether the important message alert will cause your phone to vibrate and whether the phone will notify you every single time a new message arrives (a setting I'd recommend enabling).

Note that you will need to have notifications turned on for the account in order for this to work; if you don't, Gmail will prompt you to fix that. If you'd rather not get notifications for regular ol' messages, just head back to the main settings page for your Gmail account and select the option labeled "Inbox sound & vibrate." There, you can set the general notifications to be silent -- in which case you'll see them in your status bar but won't get any audible alerts -- or you can disable them entirely by unchecking the "Label notifications" box.