Seed collecting, hybrids?

Seed collecting, hybrids?

In regards to seed collecting from plants in the garden, does the fact that a plant is labeled as an "F1 hybrid" have any bearing on wether or not the seeds collected from that plant will produce? See I was given some petunias last Spring, and they weren't "Wave" brand petunias but a similar behaving type called "Madness". On the care tag with the plant id and all it says "F1 Hybrid" and I vaguely remember reading or hearing about hybrids and seed collecting and maybe there was something about the seeds being sterile. I'm not looking for an exact replica of the parent plant, just something that possibly behaves in a similar fashion, regardless of color, spread, what have you. Just wondering if I can expect something to even grow from the seeds.
Please help me stop murdering plants!!!

Please help me stop murdering plants!!!

Another angle of the mystery bug

Another angle of the mystery bug

My room!

My room!

Still assasin bugs?

Still assasin bugs?

Jewels of Opar and Eggplant

Jewels of Opar and Eggplant

Tomato seedlings

Tomato seedlings