Thanks for the question, Nancy!
There are two approaches one can take to identify where 'connections' can be made into the "World Tree":
1) Before adding a person's profile, search for it. Pro members have access to the Geni search functionality. For Basic (free) members, you can use an Internet search engine (such as Google) to find possible profiles; with Google, for example, you can add 'site:geni.com' to restrict the search results to Geni.com.
2) Add the person's profile, and then wait a few day to see if any matches show up (e.g.: via the little circle on the Tree View, or the "Tree Matches" item in your 'merge center' ... which shows up under the little number in the upper right by your name (even if it is zero).
*** A KEY POINT *** Do NOT assume every (or even any!) 'possible' match is good.
Before proposing a merge, check out dates (if present) and preferably a couple of generations in either direction (ancestors and descendants) to see if the "tree structure" seems to be consistent. Be sure to read the "About" descriptions (if present) for some of the surrounding profiles as well (children are sometimes mentioned in the "About" of their parents).
Once you are pretty confident in a 'match' being correct, the first thing to do is ensure that the profile you added -- assuming it is of a deceased person -- is set to be "Public". (Check the "Actions" button selections.)
If the two profiles are shown in a 'match' list, the easiest is to just click the "compare profiles", which takes you to a 'merge' screen, and then click the "Yes" button.
Another way to get to that 'merge' screen is to 'move' a profile in "Tree View" and then 'drop' it onto the matching profile. (The "Move" option is on the sub-menu of the word "more" on the profile's box in Tree View.)
There are other methods as well, but those are probably the most commonly used.
By the way; using the "Move" in the "Tree View" is also the way you can take an existing profile and 'connect' it as the parent, or child, or sibling of another profile. When you 'drop' a profile onto another one, you will get a menu that asks just what relationship the two profiles have: parent, sibling, child, or the "same person".
What happens at once a merge is 'proposed' can get pretty complicated, depending on a variety of factors! But at least that gets a 'likely match' identified.
One other 'tip' I'll mention: If you find a "Public" profile that seems to match one of yours, but you are not real sure ... or you can't figure out what to do next, you can start a Discussion under that Public profile (view the full profile, click on the Discussions tab, and start a discussion with your questions ... or add to an existing one, if there are any).
One other thing: You can't "replace" an existing profile with one of yours; you can only merge them. A 'completed merge' cannot (currently) be reversed (only disconnected and re-created); however, a proposed merge *can* be rejected. That current lack of a 'merge undo' is why I emphasize making sure of a match before proposing a merge.
Phew: That's probably more than you wanted ... but at least I hope it is clear enough.
I like to think of Geni's World Tree as a giant wall of "sticky notes" with strings connecting them.
I can add my 'sticky note' of info "into the pile" that already exists for a persons, but if someone comes along later and says "here's a document that shows why some of that info is incorrect" they can shuffle around the "pile of notes" to have the best-known-to-be-correct information showing on the top.
So, in my looking around on the wall I find a 'sticky note' that seems to be the same as one I saw before ... so I can pick that first one and "move" it on top of the other one -- and all the connections between the notes follow right along (unless I explicitly then disconnect one after moving the "sticky note".
That's my "visual analogy" for how profiles and merging works.
It's not "my" tree ... or "your" tree: it's *our* tree.