Sometimes you need to just clear out leads and you have 3 options.
If you are in Customer Service version you will get a "close option", this will require you to add a note to the ticket section in the details panel of your sidebar before you can close and hide the ticket.
In the sales version, you will only see "Delete" and "block". Deleting a lead will delete all the histroy of that lead and remove it from your list of leads. If that lead contacts you again, then it will show once more, but the prior conversation will not show.
If you "Block" a lead, it will delete the lead, but it will also stop that lead from showing in your service again, this is good for spam or newsletters you may signup for.