No question, the best advice is a licensed electrician. Notwithstanding, here is how I understand the OP's issue.
He has a 120 volt welder. When he plugs it into a "normal" outlet (15 ampere) it "blows the fuse" (i.e., trips the overcurrent device). So, he has determined that a higher capacity, 20 ampere, overcurrent device (breaker) should be installed on the circuit and the wire from the breaker box to the outlet should be sized accordingly, say 10 ga copper.
Inasmuch as the circuit is 120 v, 10-2 with separate ground is reasonable. Normal configuration would be two insulated solid conductors, one with white insulation and one with black insulation, and a bare copper ground wire.
For some reason, he bought/was sold 10-2 red/black with bare ground. There would be less confusion if the OP simply would buy a length of the std configuration cable, which is not expensive for a short length. Either return the other, sell it on Craig's list, or frame it and chalk it up to a learning experience.
With the correct cable, in the box: black to new 120v 20A breaker, white to neutral bar (where there are other whites) and bare to neutral bar where other bare wires are.
Now, route the cable (properly) to the outlet. This should be a single outlet in a work box, flush or surface. The outlet should be NEMA 5-20R (see attached jpg) with the black wire attached to the darker/copper colored terminal, the white to the lighter/silver colored terminal, and the bare wire to the green terminal (and the workbox grounding screw if it's a metal workbox).
Because this 5-20R can accommodate either a 20A (i.e. NEMA 5-20P) or an ordinary 15A plug, no need to do anything to the welder plug (assuming it is in good shape).
I believe the OP more or less knew what had to be done (new, bigger breaker and cable to the receptacle) but didn't/doesn't quite understand how to do it and got further confused by acquiring the "wrong" wire. Perhaps as others have said, the "wrong" wire could be used my using the red as white and marking appropriately at each end...but, why bother?
What was unsaid, above, is the need to replace the receptacle in the workbox to the NEMA 5-20R (see jpg).
That said, get an electrician.