What do you think objectivity is, and what do you think of it?

My definition of "objectivity" is "the approach towards a philosophical matter that seeks to minimise the role of the subject in said matter".

For example:

If I say "two plus two equals four", I'm being objective. My statement should be true regardless of who is saying it, who's doing the maths, etc.
If I say "In my opinion, green apples are great", I'm not being objective. I'm being subjective: I'm acknowledging that the statement "green apples are great" is accurate for one subject (me), but it might not be accurate for other subjects (perhaps you don't like green apples).

what do you think of it?

Truth is objective and should be handled objectively. Gravity doesn't stop working because you're in a bad mood; 2+2 doesn't fluctuate between 3 and 5 depending on the observer; either a past event happened, or it didn't.

Other philosophical matters are better handled subjectively. For example, morality; something can be good or bad depending on the subject, and there's no way to handle this objectively.

If I say "two plus two equals four", I'm being objective. My statement should be true regardless of who is saying it, who's doing the maths, etc.
Even this is quite subjective, as it builds on the (subjective) acceptance of axioms. To most reading this, they would've been educated using the 8 Zermelo-Fraenkel (ZF) axioms, with the controversial 9th axiom of choice.
- I disagree that this is subjective. Even if someone hypothetically doesn't accept the ZF[C], the statement still accurately describes reality, in a way that doesn't depend on the subject. For example, you can't start with two apples and two oranges and have five or tree fruits.

Being able to take a step back and see subjects, siuations and problems as what they are, based on facts, proofs and truths, unfettered by personal experience, opinions or anecdote.

I think it's a worthwhile pursuit, but I also accept that we often fail to be truly objective for reasons of expediency, motive or more commonly, emotion.

I don't think there is such a thing, but there are degrees from subjective to objective.