"Conspiracy" is an agreement between at least two people to undertake some act, and generally refers to an agreement to undertake a criminal act. By definition, then, there is no such thing as a lone conspirator in the same way that there is no such thing as an unmarried wife.
U wants to kill people with bombs sent through the mail. If B agrees to help, U and B are co-conspirators. Iannelli v. United States, 420 U.S. 770 (1975) ("The essence of the crime of conspiracy is agreement.") The very act of agreeing to undertake the crime completes the separate crime of conspiracy, so U and B are correctly labeled co-conspirators.
If U does not recruit B, he is just a guy with an idea, and he has not committed any crime. If he begins to plan to commit the crime, he has still not committed a crime. If he takes a substantial step toward completing the crime, he has committed the offense of attempted murder. United States v. Resendiz-Ponce, 549 U.S. 102 (2007) ("[T]he mere intent to violate a federal criminal statute is not punishable as an attempt unless it is also accompanied by significant conduct.") If he succeeds in carrying out the plan, he has committed murder and is simply called a murderer.
"Civil conspiracy" is not a crime; it is an agreement between at least two people to break the law. That could be an agreement to break the law by committing a crime (e.g., conspiring to kill the president) or it could be an agreement to break the law by committing a civil offense (e.g., publishing defamatory statements).