Latitude D620 -> 64-bit or 32-bit?

If the processor is a Core 2 Duo, you can use 64-bit.  If it's a Core Duo, Core Solo or Celeron, 32-bit only.

Short answer: Yes, you can.

Long answer: You can install a 64-bit Operating System just fine on this model. Dell offers support for Windows Vista 64-bit as well as Windows XP 32-bit.

Bear in mind however that if you plan on using a different OS (especially x64) then you are going to run into irritating driver issues - some Dell drivers will work, others will not.

At this point, make sure the BIOS is updated to version A10 - http://www.dell.com/support/drivers/us/en/19/driverdetails?driverid=R186796

For a Wifi (wireless LAN) driver to get you started in case of trouble, you'll want one of these:
 
http://www.intel.com/support/wireless/wtech/proset-ws/sb/CS-034041.htm (Covers Windows XP through to 8.1, both x86 & x64)

Once you've been through all applicable drivers and Firmware from the Dell Product Support page, located here:

http://www.dell.com/support/home/us/en/19/product-support/product/latitude-d620/drivers

Then download SlimDrivers, a free cloud-based Driver utility (unless you have a preferred, paid alternative) here: https://www.slimwareutilities.com/slimdrivers.php

It's not perfect, but it WILL help you get your sound going, among other things

I would think your biggest challenges will be drivers for higher resolution graphics and your chipset. While Windows 7 has some decent compatibility options, don't assume you'll get overly far with them.

^^^^^^^^^^^^ NOTE: click on "Download Now from SlimWare Utilities" instead of from DOWNLOAD.COM; sites like that, Softpedia, FileHippo and Brothersoft are full of Browser Hijackers and other bundles of joy depending on your luck.

In addition, though a 64-bit Windows XP can run comfortably on the stock 2GB of RAM (as I have experienced first hand), you might consider adding some memory for Windows 7. While Starter and Home Basic Editions might coast, other Editions will have more of a hard time coping, especially after all applicable Windows Updates and installed applications.

The D620 takes DDR2 SODIMM memory, and offers two slots that support 200-pin DIMMs. Understand that, although you can shove in up to 8GB (2x4), the cost after 4GB (2x2) becomes very high (remember the memory requirements in that Era). Hynix offers a well priced and apparently very capable solution (HYMP125S64CP8-S6). Right now Amazon for example has them for around 25$ each stick, brand new.

Remember, if you are running an x86 OS and you find that nothing x64 will function, that is a limitation of your software, and not necessarily a limitation of your hardware.

Thx a lot!