The phrase public enemy number one, or public enemy no. 1, denotes that which poses the greatest threat to the welfare or security of a community, nation, etc.

Its primary meaning is the first person named on a list of wanted criminals, i.e. the person regarded as the one it is most important to apprehend.

The phrase was first used, in the USA, to designate the American gangster Al Capone (Alphonse Gabriel Capone – 1899-1947). On 17^th September 1930, the Chicago Daily Tribune published a list of twenty-six public enemies with these explanations:

Every important gang leader in the city was included in a batch of 26 vagrancy warrants issued last evening by Municipal Judge John H. Lyle in the Felony court. The list comprised the “public enemies” named some months ago by the Chicago crime commission.
[…]
The revival of the old vagrancy law was brought about several weeks ago by Special Assistant State’s Attorney Charles Rathbun, who has been assigned to investigate the slaying of Alfred Lingle. Prosecutor Rathbun and Chief Investigator Pat Roche used the vagrancy law with effect on Israel [Izzy] Alderman and Joseph Condi and this started the present campaign.

the beginning of the list, as published by this newspaper:

                                                                           List of “Public Foes.”
The complete list of those named in the warrants, as they are designated by the judge in the warrants and by the crime commission in its list of the “public enemies,” follows:
ALPHONSE CAPONE, ALIAS SCARFACE, known as the chief of gangland.
RALPH CAPONE, brother of Alphonse, and chief lieutenant in the traffic of vice, gambling and liquor. He is now under conviction by the government for income tax evasion.
TONY VOLPE, ALIAS MOPS VOLPE, one of Capone’s chief bodyguards, who specialized in handling ganglands dog track ventures.

The earliest instance of public enemy number one that I have found is from the Albuquerque Journal (Albuquerque, New Mexico) of 10^th October 1930:

AL CAPONE’S BROTHER WINS HIS FREEDOM
‘Bottles’ Is Released on Bail; Was Held in Chicago Jail Cell During Afternoon, However

Chicago, Oct. 9 (UP)—Ralph “Bottles” Capone, public enemy number 2 on the Chicago crime commission list, spent an uncomfortable day in jail before Judge John P. Lyle finally approved his bond Thursday night.
Brought into court earlier in the day, Capone asked for “courtesy” while officials wrangled about a $10,000 bond he obtained irregularly for release on vagrancy charges Wednesday.
“I give no courtesy to gangsters,” said Judge Lyle. “Take this man to jail and treat him just like any other criminal.”
So off to a cell police [sic] trundled the resplendent younger brother of “Scarface Al,” who is public enemy number one. There young Capone, erstwhile maestro of a string of night clubs and reputed head man in the illicit whisky department of the Capone crime syndicate, refused to answer questions put to him.
After he had moped all afternoon in the cell, Judge Lyle approved his bond and granted a change of venue for his trial on the vagrancy charge. If he is found guilty the dapper young Capone must serve six months on the county rock pile. He already is under three year [sic] sentence for alleged failure to pay federal income taxes, his case being under appeal.

This is the beginning of an article from the Green Bay Press-Gazette of 13^th October 1930:

CAPONE STILL PLAYING TAG WITH POLICE
‘Public Enemy No. 1’ Eludes Law During Raid In Cicero.

Chicago—(AP)—The “No. 1 public enemy,” Alphonse (Scarface) Capone, continues eluding the law despite the snares set for him.
A surprise raid on Cicero, the suburb that acquired an unenvied name as the haunt of the Capone “mob,” failed to flush the gang leader yesterday.