This could be one of several models. As you say, they all load through a hole in the side in the buttstock. All operations are through the bottom of the action. The cartridges feed over the top of the bolt, the ejection is out the bottom. The gun is charged by drawing the bolt body to the rear through the bottom of the action.

Some possible model numbers:

The 24
http://www.remington.com/library/history/firearm_models/rimfire/model_24.asp

The 241
http://www.remington.com/library/history/firearm_models/rimfire/model_241.asp

http://www.gunbroker.com/Auction/ViewItem.asp?Item=129370338

You will also see this gun from Browning and FN
Here is a Browning
http://www.gunbroker.com/Auction/ViewItem.asp?Item=129368010

Here is one from FN.
http://www.gunbroker.com/Auction/ViewItem.asp?Item=129283299

John Browning was the designer of this gun. Your make and model number will be on the left side of the action.