What tools in Premiere can I use to create the effects used in this video?

With nothing selected on the timeline(click on an empty area),

press and hold on the Rectangle Tool and choose Ellipse Tool.

with this tool selected go to the program monitor,

click and drag while holding the 'Shift'(windows)

or 'Command'(Mac) key,

this will create a perfect circle, or if you don't want a perfect circle

simply click and drag until the shape looks the way you want.

From the Effect Controls panel you can scroll down

to your shape properties and make all your adjustments

including opacityand color controls from there.

This will automatically create a graphic clip on the timeline

where the playhead is. you can ALT + drag

to make duplicates of the graphic clip and make

the other adjustments, position and scale offsets and edit cuts accordingly.

Let us know if you need more help

eikonoklastes gives you a good explanation of how to do it in Premiere Pro, but you may find it easier to do it in After Effects.

- Create a Color Matte and add it to your sequence.

- Then, with the Color Matte clip selected, in the Effect Controls panel, create an Opacity Mask and define the shape you want.

- You can then decrease the Opacity slider from 100% to the level of transparency you want.

- Duplicate this clip as many times as you need, and adjust the mask for each one.