How can I merge two photos using software that is provided with OS X? I have two photos of a group of ten people. Unfortunately, one person has the eyes closed in each photo. 

I would like to merge one person from one photo into the other so that everyone's eyes are open in one picture.

Are there any tools or applications that are provided with OS X by default that will allow me to do this?
You don't need anything special at all, you already have the tools you need to give it a go - frankly it's worth a pop in Preview.

I'll have a quick and dirty go in this answer and see how we go.  I've taken 2 photos, one with my eyes open, and one with them shut.  I will transfer the eyes from one to the other to create a composite picture of the two.  Excuse the poor quality, but they are taken from my webcam in a dimly lit room:

Eyes Open:


Eyes Shut:


Opening the picture I want to copy from (just double clicking it should use Preview as the default app on most systems) ensure first that you have the correct edit tool bar showing, and select the elliptical select tool:




Now, draw a nice oval around one eye, starting in the top left and moving right and down.  If you get it wrong, just single click anywhere outside the selection and start again.  Repeat until you have got a good outline of the eye.  You can fine tune by grabbing the handles on the selection tool and moving them a little:



Now simply copy this selection, open up the other picture, and paste it in, moving it to the required position (and potentially resizing a tiny bit if required).  To remove the selection marks, click anywhere outside them to "set" the paste.  If it's not in the right position, click undo, and paste again:



Repeat with the other eyes, and here is a finished composite:



Top tips:


This works better with smaller snippets, obviously the more closeup it is, the more accurate your cropping has to be and the more obvious the effect is.
Your photos need to be pretty similar in terms of lighting, zoom levels etc to get a consistent look.
Zoom in if required to get more accurate selections.
If it's a group shot as you say, then chances are everyone's face is quite small unlike my full face image example, and any imperfections are less likely to be spotted
Keep your selections as small as possible, copying a couple of eyes is going to be easier and look better than trying to move a whole head, or person.
Other software will provide for more complex features like smoothing the joins over to reduce any hard lines where the colouring is marginally different etc
If you had to copy say a piece of eyebrow in order to get the whole eye, and it will not line up in the new picture, then use an intermediate empty picture file, paste it in, cut out the bit you don't want, re-copy it and carry on.