Welcome, initially this is all going to sound confusing but I promise after a short while of handling Lego it'll all make sense.
So I'll break down the terms for you
Element ID - This is a unique number to a part, in a certain color, with a certain print if it has one. For example a 2 x 4 brick has a separate element ID for every color and every print. This number is the one Lego print on the inventory lists in manuals or on boxes.
Design ID - This is more complicated. So every mold of a part has the same Design ID, it is usually found molded into the part somewhere. Where confusion comes in is for Printed parts. For example a 2 x 2 Brick has the design ID 3003. This is molded into the underside on the brick. BUT if that 2 x 2 Brick has a print on it Lego assign it a new Design ID. For example 3003pr0038 (which is our number designation, more on that later) That is a 3003, 2 x 2 Brick and will have that number molded on it but Lego gave it Design ID 25660. Another complication with Design ID is when Lego slightly change how they make the part. If the mold is slightly different, or they change a small detail about the part, they will assign it a new Design ID. To make things easier for Lego builders most websites (including Rebrickable) combine these parts. On Rebrickable, if the change is considerable a new part is created and a mold relationship is attached to the part so users can find similar parts. This is why you'll see multiple Design ID's connected to the same part.
BOID - This is BrickOwls unique inventory number. It has no more meaning than that. Only BrickOwl use that number. It means Brick Owl ID
Rebrickable numbers - We create part numbers for our inventorying needs. Our numbers usually follow the first know Design ID, which is commonly the one molded on the part. Then when that part gets prints we use the designation "pr" for Print, "pat" for Patterns (multicolored marbled parts), or "c0x" for pre-assembled parts, like a Torso, Arms, and Hands part.
In my experience Lego.com isn't very helpful. Rebrickable is usually a great place to start when trying to find a part (admitting bias). We link to all other sites too once it's found. Our Search tool is quite comprehensive and once you get more familiar with Part categorizing it becomes extremely easy to find a part with no information.
The easiest way to find a part is with element ID. That gets you to the exact part Lego intended.
Posting here for help on part identification is also useful as our users are usually more than happy to help.
Hope that helps a little. Good Luck.