Use the V key to access your Digivice. On the lower half of the menu you should see two item slots. These are your Chipset slots. Simply drag-and-drop the Chipset from your inventory to the slot to equip it.
You already know what the attributes are, but as for the item icon and the abbreviation at the beginning, that effects which "family" of Digimon is effected. Aside from the Data/Virus/Vaccine and Elements, these are also Families and each Digimon has at least one, based on their overall design.
WG = Wind Guardians, usually birds or other air-themed Digimon.
DS = Deep Savers, Fish or aquatic Digimon.
DR = Dragon's Roar, Dragons obviously.
NSp = Nature Spirits, woodland creatures and plants.
NSo = Nightmare Soldiers, demons and other evil monsters.
JT = Jungle Troopers, Bugs and other jungle-native animals.
ME = Metal Empire, robots and other Digimon characterized by metal.
VB = Virus Busters, Angelic or heroic Digimon.
DA = Dark Area, I'm not 100% on this one but it seems reserved for the most evil of Digimon.
I may have missed one or two, there are quite a few and they aren't easy to remember.
If you press D to open your Digimon status you can see a number of icons underneath your EXP bar. Among those icons will be at least one circular icon with one of the above abbreviations in it. This is how you tell what Family your Digimon is from. You can also open your Dgivolution menu, mouse of the Digimon in question, and get the information that way. So, for example...
I have a Falcomon. Being a bird, he's a Wind Guardian, so I equip a WG Chipset. I also have a Fanbeemon, who is also a WG but evolves into Waspmon, who is WG and ME. If I equip a ME Chipset he will gain the benefits of both, but only while he is evolved.
Now, as far as the Digi-Eggs, I'm not sure. But I think if you scan them you'll get a Chipset from that Family.