In eukaryotic organisms, photosynthesis takes place in the chloroplast. These cell organelles consist of three membranes. The innermost membrane forms stack-like structures called thylakoids. More specifically, photosynthesis takes place in the thylakoid membrane.
Figure 1: Structure of a chloroplast.
There is an interesting story to the two surrounding membranes: Chloroplasts used to be prokaryotic algae that were ingested by early eukaryotic cells. Instead of being digested, they turned out to be useful in providing energy for the cell. The two symbiotic cells became dependent on each other, and the prokaryote became what we now call an organelle. The two membranes are still remnant from the vacuole of the eukaryote (outer membrane) and the plasma membrane of the prokaryote (inner membrane).