What’s a Safe Way to Kill a Weed?

Since Spring is full swing, I have planted tomato and pepper plants, a few flowers, pressure washed the deck and cut the grass a few times.

I try to keep my yard looking like someone actually lives here. I snapped the picture above this afternoon. It's a wretched weed of some kind. What good does it do to try and keep your yard in good shape when these things are ALL over your yard?I have mowed them down several times but they spring back bigger and badder then before.

I know a little about a lot of things but I know nothing when it comes to what chemicals to put on your yard to get rid of weeds. I am scared I will kill the grass. I don't want burnt spots all over the yard!

What can I put on these ugly weeds to kill them dead but leave the grass?