Any time a new bond is formed, take a look at the atom that's accepting the lone pair of electrons. Does it have a full shell of electrons? Oftentimes it does. If so, that means you're going to have to break a bond that's attached to that atom. If it's a single bond you're breaking, what you end up creating is called a leaving group.
Note how the arrows operate. The arrow always shows a pair of electrons going toward the leaving group. That means the charge on it is going to become more negative by 1 when it leaves. So if the leaving group is positively charged, it will become neutral, and if it's neutral, it will become negative.
The identity of the leaving group is crucial to whether the reaction will happen at all. There are "good" leaving groups and there are "bad" leaving groups. Reactions are more likely to take place when you can displace a good leaving group.
What makes a leaving group "good" or "bad"? Thankfully, there's one simplifying factor to look at when deciding this: its basicity. Good leaving groups are weak bases.
How do we know what are weak bases? There's a useful tool for that - it's called a pKa table. Many pKa tables only specifically give you the identity of the acid, but if you think about it, it also gives you information about the conjugate base of each acid. The conjugate base is the part left over when you lose H+. The stronger the acid, the weaker the conjugate base.
And the weaker the conjugate base, the better the leaving group. So a pKa table is a great guide to leaving group ability.
One word of caution: pKa measures an equilibrium, whereas leaving group ability is based on reaction rates. So although the correlation is very good, it isn't perfect.
Here's a pKa table, annotated.
The trend is pretty clear - in general, the weaker the base, the better the leaving group. Furthermore, note how we (almost) never see alkanes or hydrogens as leaving groups. That's because they're strongly basic anions - and very unstable.
You might note that I have carefully avoided discussing fluorine. Fluorine tends to be a very poor leaving group for SN1/SN2/E1/E2 reactions. you may see some examples where F can act as a leaving group when it is attached to a carbonyl carbon or an aromatic ring. These reactions (addition-elimination reactions) are a little bit different in that the rate determining step is not so related to loss of the leaving group. There are some extra factors at work in these situations that we can discuss if you're curious.