The is-ought gap noticed by Hume is just the fact that any valid argument for
an ought claim must have at least one ought claim among its premises. Ethical
claims about what is right or wrong are ought claims. So a valid argument for an
ethical claim must have some other ethical claim among its premises. No number
of facts about how things are deductively entails that they ought to be one way
or another. For instance, the fact that people drive drunk doesn't entail that
driving drunk is right.
Consider this argment
The fetus is a person.
Abortion is the killing of a fetus.
Killing persons is wrong.
So, Abortion is wrong.
This valid argument has an ought claim as a conclusion and an ought claim among its premises. Remove the ought claim from among the premises and the argument is no longer valid. No replacement of the 3rd premise with an is claim can make it valid.
Hume was a strict empiricist, which means he thought all of our knowledge is ultimately based on sense experience. He also says we have no sense impressions of wrongness and rightness. Finally, Hume observes that ethical "ought" claims are not logically entailed by any of the “is” claims we do know through sense experience (this is the is/ought gap). So, we have no empirically significant concept of wrongness or rightness and hence no knowledge of ethical truths. It takes a bit more argument to reach the conclusion that there are no ethical truths. But this is enough for our purposes. So, that is Hume's subjectivism in a nutshell.
[an aside: Lots of new students think the view that there are no ethical truths means that anything goes or anything is OK. But "anything is OK" is an ethical claim. If there are no ethical truths, then this is not a truth either. To deny that there are ethical truths is not to deny that there are moral sentiments. So whether or not subjectivism is true, people will be angry with you if you lie and steal, your parents will still send you to rehab if you do meth, etc. Subjectivism doesn't mean that people will or should act or feel any differently when people break the rules.]
An ethical realist can respond to Hume's subjectivism in one of two ways: Reject Hume's empiricism or argue that we do have empirical experience of ethical truths. I think both avenues are promising. Hume empiricism has other problems it may not survive anyway. We have no sense experience of causal connections either, but we aren't about to stop believing in causation, for instance. And I rather like the view that we experience moral properties through our moral sentiments. You are experiencing something when you feel indignant after being treated unfairly. Couldn't this just be an aspect of your sense of your own self worth?
Subjectivism faces a challenge in explaining how to understand our ethical discourse assuming our ethical claims do not express truths or falsehoods. We appear to reason about ethical issues, but this can't be what we are doing if there are no ethical truths or falsehoods. I discuss this problem in Mod 2 in "against subjectivism"
The is/ought gap is just Hume's observation that no claim about how things ought to be follows validly just from claims about the way things are as a matter of fact. For instance, no collection of data about literacy levels deductively entails that we ought to spend more on education. However, data about literacy levels in conjunction with the right ought claims (for instance, "everyone ought to be able to read") can deductively entail that we ought to spend more on education.
Hume's is/ought gap by itself does not entail that subjectivism is true. It is just an observation about what valid arguments for ethical claims requires (namely, that they require other ethical claims among the premises). But re-examine the discussion of Hume's subjectivism above. Not that subjectivism is the view Hume reaches from recognition of the is/ought gap AND his strict empiricism AND denying that we know any ought claims through emprical experience. It takes a good deal more than the is/ought gap to generate a valid argument for subjectivism.