Our mystery seeds from last week are from a cucumber, Cucumis sativus.
Cucumbers grow on a vine.
They have separate male and female flowers on the same plant. In order to produce fruit, either an insect pollinator or a human must move the pollen from the male flower to the female flower.
You may not have recognized the seeds from what you have observed when you cut open a fruit to eat. That is because we eat the fruit before it is fully ripe and the seeds are not yet mature.
Here the seeds are soft and not fully filled.
The mature seed has a hard seed coat.
If you are interested in saving seeds, you should leave the last of the season cucumbers on the vine to ripen fully. Often the fruit will yellow as it becomes fully ripe. Apparently cucumber seeds also benefit from a period of fermentation, like tomato seeds do. It has to do with the gel covering inhibiting germination.
Although not the best quality, this video explains how to do it (there is a pop-up ad):
Cucumbers are interesting plants. Scientists have found that cucumber plants produce chemicals from their roots (while they are fruiting) that inhibit the growth of other nearby plants. This has implications for people trying to grow plants hydroponically, as the chemicals enter the water.
Jing Quan Yu and Yoshihisa Matsui. (1994.) Phytotoxic substances in root exudates of cucumber (Cucumis sativus L.) Journal of Chemical Ecology. Volume 20, Number 1 (1994), pages 21-31.