What are the differences between these groups polled by FiveThirtyEight?

"All adults" means that those people who can't vote are included, this tends to make the poll less accurate and skews towards the Democrats.

"Registered voters", this means that voter registration lists are used in the sampling process. (In the US, registration is an "opt-in" process, and varies from state to state. To vote you must have a current registration at your precinct) All registered voters are eligible to be sampled, including those who don't vote. Again this tends to cause the poll to skew Democrat by a point or two.

Polls of "likely voters" use sampling techniques that try to fit the sample to the population of those who actually are likely to vote in an election. The voting population is older, whiter and more Republican than the US adult registered voting population. These polls are more predictive but intentionally omit people who the pollster believes (based on historical evidence) are less likely to vote. This uses census and exit poll data, and can skew the results if an election doesn't follow historical norms.

It seems that polls of "Voters" is a PPP term. Unlike other pollsters they just ask people to hang up if they don't (or won't) vote. This doesn't depend on census data, but does depend on people being self-aware of their likelihood of voting and not dissembling to the automated caller that PPP uses. Nevertheless, PPP gets comparably accurate results as other pollster with this methodology.

In general, if you want know "how America feels" about an issue, then polls of all adults are best. If you want to predict "who will win" then polls of likely voters are best (with polls of registered voters an acceptable second place early in the cycle) Most firms don't switch to LV until Labor Day. Likely Voter screens can increase accuracy on head-to-head matchups close to election day, but can also introduce a different set of biases; "house effects" are typically the result of a particular set of criteria for how likely a person is to vote, whereas polls of all adult and registered voter groups don't suffer bad assumptions about the likeliness of the voter to vote.

Every pollster will have their own terminology, and because there aren't any polls that have breakouts for "Adults" and "Voters", it's hard to tell what the difference is (if there even is one).

As for the other two, both of these will be subsets of Voters/Adults, but it's not necessarily the case that "Likely Voters" will be a subset of "Registered Voters". In several states, it is possible to register to vote on election say itself, so it is possible that some voters will say they are likely to vote, but haven't registered yet.