Welcome to Pathfinder! I'll quote the rule, then break it down and explain it.
Each domain grants a number of domain powers, dependent upon the level of the cleric, as well as a number of bonus spells. A cleric gains one domain spell slot for each level of cleric spell she can cast, from 1st on up. Each day, a cleric can prepare one of the spells from her two domains in that slot. If a domain spell is not on the cleric spell list, a cleric can prepare it only in her domain spell slot. Domain spells cannot be used to cast spells spontaneously.
In addition, a cleric gains the listed powers from both of her domains, if she is of a high enough level. Unless otherwise noted, using a domain power is a standard action. Cleric domains are listed at the end of this class entry.
A cleric gains one domain spell slot for each level of cleric spell she can cast, from 1st on up. Each day, a cleric can prepare one of the spells from her two domains in that slot.
As you noted, the table of spells per day has a "domain slot" +1 indicated on it. For each spell level you have, you can prepare your base number of spells (1-4), plus how ever many you get from having a high Wisdom score, plus a single domain spell chosen from the two that are available to you.
So, for example, as a 2nd level cleric you have
2+1 1st level spells each day. This means you can prepare any two spells you want (really three, since you probably have a bonus spell from Wisdom, or four if you have a 20+ Wisdom). In addition to those two (or 3, or 4) spells, you can choose to prepare either burning hands from your Fire domain or cure light wounds from your Healing domain. You can't prepare both in that single slot - you need to choose between them.
When you get access to 2nd level spells (confusingly, at 3rd character level), you'll get to choose between produce flame and cure moderate wounds for that 2nd level slot, in addition to continuing to choose between burning hands and cure light wounds for your 1st level slot.
If a domain spell is not on the cleric spell list, a cleric can prepare it only in her domain spell slot. Domain spells cannot be used to cast spells spontaneously.
This means that since burning hands is not normally a cleric spell, you can only prepare it in that domain slot. Cure light wounds, on the other hand, can be prepared in any regular slot you have in addition to being able to prepare it in the domain slot. (Or spontaneously cast, but that's a separate issue I'll mention later.)
In addition, a cleric gains the listed powers from both of her domains, if she is of a high enough level.
This means that unlike spells (where you have to choose), you get the full set of powers from both domains, and can use them each separately their specified number of times per day.
As for spontaneously casting - a cleric can normally turn any spell they have prepared into the appropriate level's cure spell. That means that those two (or 3, or 4) regular spells can all become cure light wounds as needed. However, your domain slot cannot be used that way. If you prepare burning hands, you can't convert it to cure light wounds. That's why you might want to choose to prepare CLW from your Healing domain, just to get that extra cure.