Partners on VDex will aid you in specific ways. There are a total of 12 partners, but two of them (Hari and Nao) will join you at the same time. Each VDex partner provides specific aid in one area or another. Every new user will start with Yun, who will give you a discount on items purchased in shops, which is especially useful for new members, as the first week will see you almost constantly struggling to keep funds.
As well as providing specific benefits, partners have a happiness value. The more a partner is happy with you, the better their aid to you will be. Once the partner is maxed out on happiness, an NPC in Pallet Town will give you a special item. These special items can be exchanged for an item that summons a legendary Pokemon, by an NPC in Sinnoh's Fight Area.
To find out which partner is right for you, do some research before switching partners: http://forum.vdexproject.net/viewtopic.php?p=25621#p25621