I’m pretty sure I never learned this term, if there is one. Anyway. . .

For example:

X[sup]2[/sup] + x + 1 = 0

The “x squared” component is called the. . . what? Is there a name for the squared, cubed, (and higher powers) components (other than, in this case, the “squared component)?

The names I’ve usually heard are:
[ul][]x[sup]2[/sup]: Quadratic[]x[sup]3[/sup]: Cubic[]x[sup]4[/sup]: Quartic[]x[sup]5[/sup]: Quintic[/ul]Beyond this, I’ve not heard any names in common usage, though that doesn’t mean they don’t exist.

Also, I would call the example in the OP a “2nd order polynomial”. Often, polynomials are used to approximate a function (by expanding that function out as an infinite series). If we were to truncate it to: a +bx, then that would be the “first order approximation”, since we only took it out to one power of x. That is what mathematicians and scientist mean when they use that term (1st, 2nd etc order approximation). Sometimes they will used “first order” to mean just an approximation, if they are talking in the vernacular, but if they are being rigorous about things (like in a paper, or a talk to other scientists), they will mean the order of the polynomial expansion they are assuming in their model.

To add on
2nd power = quardratic
3rd power = cubic
4th power = quartic or biquadratic
5th power = quintic
But why doesn’t it go higher to sextic, septic, octic etc.? I think it is because a lot of the work in algebra in Western Europe a few centuries ago dealt with finding formulae that would give the roots of various degrees of polynomials using only field operators. The formulae for quadratic, cubic and quartic were found and so next in line was the quintic formula. Abel proved a general form for the quintic could not be found (which Galois improved upon) and so the naming of polynomials for higher degrees stopped.