For my Acer Aspire AX3810-E3852A desk top, I want to know if I can get a new sound card that fits within the confined area of its small case.

If you have an available, (Open. Unused), PCI slot, or PCI x1 slot, Yes is the answer.

(Not PCI Express x16 slot. This slot is reserved for graphics cards only)

From what little information that I can find on your exact Aspire Model Number,

http://shop2.aol.ca/shop/product--catId_1001610__locale_en__productId_9024282.html

,you should have an available PCI slot, and perhaps you have a PCI x1 slot, and it is available.

(According to the simple specifications I just read, I KNOW you have a PCI X16 slot for a graphics card, and it IS available.

=>IF, you did not install a graphics card already. If you are using the Acer the way it came from the factory, you are using Integrated Graphics. Intel GMA X4500.
The PCI Express x16 slot is open, and available)

I need to know if you are aware, what the various expansion slots look like. Do you know what a PCI slot looks like? A PCI x1 slot? A PCI x16 slot?

A PCI slot is always White in color. Simple ID there.
A PCI x1 slot has 36 contact pins in the slot. 18 on each side.
A PCI Express x16 slot has 164 contact pins. 82 on each side.

More information,

http://en.wikipedia.org/wiki/PCI_Express

Forgive the above if you are computer sav vy, and are aware of the various expansion slots. I'm just trying to give you all the information I can.

Your computer is an SFF style computer. Small Form Factor. You need a Low Profile sound card. The profile of the card is low in height.

These are examples of a low profile sound card. It is not an advertisement for the website, nor the manufacturer.

http://www.newegg.com/Product/ProductList.aspx?Submit=ENE&DEPA=0&Order=BESTMATCH&Description=low+profile+sound+card

(5.1 Channel is pretty decent. 7.1 Channel is Really decent)

This would depend on what type of motherboard you have. Desktop motherboards can have a variety of slots to install many different types of cards. Most comman slots could be for AGP, PCI and PC Express type cards. Need to get board specifications to know what your board will except. If you don't have the specs, you can locate them online from the motherboard manufacturer.

Ya u can check if u have a pci expressed x1 slot it is small slot along the pci slots...