802.1q is the technical standard for VLAN tagging (http://en.wikipedia.org/wiki/IEEE_802.1Q). This standard includes the placement of a "VLAN Tag" inside the header of Ethernet frames. This tag allows a link to carry multiple VLANs, as long as both devices recognize 802.1q tagging, because the tag contains the VLAN ID that the traffic belongs to.
Such a link is commonly referred to as a "trunk link" or "802.1q trunk", etc. In such an environment, there is typically a single VLAN that's assigned the role of "native VLAN". The term may also be "untagged VLAN", because that's what a native VLAN is - a particular VLAN that crosses a trunk link with no VLAN tag. Since there's no way to identify which VLAN a packet belongs to without this tag, only one VLAN can be designated the "native VLAN", and it's a good idea to make sure this value matches on both sides of a trunk.
In ESXi, you can define port groups with a VLAN ID. If you leave this field blank (or specify 0) then that port group's traffic egresses the host with no VLAN tag. This is okay if you have only one port group on the host and that vSwitch only plugs into one link, because you can simply make that port an access port, no trunk necessary. However, you need multiple VLANs to be passed on the same link (or link bundle), so I would make sure trunking is configured on the switchports that your host connects to, then all you'll need to do is enter the appropriate VLAN ID per vSwitch port group. ESXi will tag frames entering that port group as they exit the host.
It's important to configure your switchports in "VLAN Trunk" or "VLAN Tagging" mode because non trunk ports do not accept tagged frames (they get dropped). Trunk ports accept tagged frames, and you will be sending tagged frames from your ESXi host with the above configuration.
From what I can tell, the switch you show in the diagram should support all of this but it might not be the most intuitive or use the same terms. I'd recommend sticking with the documentation and see if you can get it working.