So what is aerospace grade aluminum? Aerospace grade and regular aluminum or both aluminum. Do they have a different composition or strength? What makes one aerospace grade?
Read David Hammen's comments below the opening question. once you get past the (well deserved) sarcastic humor he absolutely nails the answer. There is no checklist for getting an alloy of anything to be called aerospace grade. There is no sign like at Disneyland saying "your alloy must have at least XX strength to weight ratio to be aerospace grade". There is no rule that says how precisely controlled any additives or process must be (other than the spec for that material) to be "aerospace".
edit by BradV: there is a small caveat required here. From time to time a specific project might discover that an existing well defined alloy performs more to their needs if they more closely control one or more constituent alloying elements. This might take the form of specifying instead of standard nickel content of 5% +/- 2% we want nickel at 6% +/- 1% and we are willing to pay the price for this tailoring. We will buy this material to our specification XXX-YYYY.
All modern materials are specified in standards such as ASTM, ASME, ASE, Mil-Stds, MS standards. Anyone can call out any material on an engineering specification/drawing. Just because a material has a specification does not mean you actually GET that exact material.
The trick is to make sure you get the exactly right material properties today, next week, next year, in 10 years. Unless you procure from a high quality certified provider you are likely to get what purports to be the same material but can vary widely from nominal values. Straying widely from nominal can destroy aircraft/spaceships/satellites.
You might design/build/test/flight-qualify an aircraft this year. Next year the production build begins and the materials might be bought from a different mill or even a different country. In ten years you are still building more and more of the same aircraft. You must be confident that the most recent one has the same structural integrity as the 1st.
Strict process controls, rigorous sampling, willingness to throw out a bad batch, documentation and certification of the whole material chain from ore to smelter to mill billet to local distributer is not cheap. But it is worth it for high risk endeavors, like making sure your airplanes don't fall apart.
While it has been implicitly explained in other answers, I don't think it has been spelled out explicitly yet: what makes aluminium "aerospace-grade" has nothing to do with the aluminium itself.
"Aerospace-grade" is a trait of the supplier and its processes, not of the material being supplied.
"Aerospace grade" is a marketing term that has no meaning.
I think that the question and the other answers so far may have meant "What makes aluminium MilSpec". Which means, to summarise the other answers in a single sentence "you pay someone else to test your inputs and suppliers rather than testing your inputs and suppliers yourself".
Whether you do tests yourself or pay someone else, the tests are done against "MilSpec" requirements, formal requirements developed by the US and UK military.
Decades ago during study, the professor showed us some small cylinders made of Duraluminum, the diameter was about 5 mm. After heating this alloy would harden when left at room temperature for several days. The difference was incredible, we could bend the probes that were heated some hours ago easily with our hands. Bending a hardened probe was impossible. The tensile strength of Duraluminum is about 5 times and more better than that of pure aluminum.
So using pure aluminum for aircrafts and spacecrafts does not make sense. Duraluminum is a copper aluminum alloy, the alloys 2014, 2024 and 2048 are used for aerospace today.