A German immigrant ancestor has a "Registration Affidavit of Alien Enemy" on file. What does that mean exactly?

Short answer

This was a requirement for all "natives, citizens, denizens, or subjects" of the German (and, a few months later, the Austro-Hungarian) Empire; thus, 'alien enemy' registration was not about targeting individuals and - for the vast majority - did not lead to internment. It did mean, though, that thousands of people's lives were adversely affected in one or more ways.

For those interned, most were put in camps because of suspected 'disloyal behaviour' or being 'involved in radical politics or labour unrest'.

Details

Registration was required for

all males in the United States older than 14 who were "natives, citizens, denizens, or subjects" of the German Empire were required to register as alien enemies. In 1918, an act of Congress included women aged 14 and older.

In more detail,

After America entered the war on 6 April 1917, German nationals were subject to a series of restrictions on their lives, property and freedom of movement imposed by President Woodrow Wilson (1856-1924) under the 1798 Alien Enemy Act, including bans on entering "forbidden zones", compulsory registration with the police or U.S. postmasters, prohibition on owning signalling apparatus, radios and firearms, and so on. At various points these restrictions were increased or exemptions revoked, largely to appease public opinion. They were also extended to Austro-Hungarians in December 1917 and to all female enemy aliens in April 1918.

'Alien enemies' had to have a registration card on them at all times and permission was needed to travel or change residence. Restrictions were ended on 25th December, 1918, some six weeks after the armistice.

This requirement was deemed necessary because

government officials believed these denizens to be the most probable spies and saboteurs, they felt it necessary to classify and monitor these individuals.

Failure to register could have severe consequences:

The Minnesota Commission of Public Safety laid out strict consequences for all delinquents: "Any alien resident of the State of Minnesota who fails to register and make prescribed declaration on one of the above designated registration days will be interned or subjected to other action which the Minnesota Commission of Public Safety will prescribe."

For failing to register,

Internment could last for an unspecified period of time, or until the end of the war.

Nonetheless, many Germans delayed registering so long that the deadline for registration in Minnesota had to be extended by two days due to a deluge of applicants on the (original) deadline day.

It appears that some states went beyond the restrictions mandated by the Federal government:

Several Midwestern states which had previously granted suffrage to aliens holding first naturalization papers rescinded these laws. Aliens were also expelled from all ships and boats except public ferries, as well as anything deemed to have military importance, from wharves to railroad depots.

while in Wisconsin posters were put up telling aliens "Work or Go to Jail", the penalty for 'listlessness' being 3 months in jail.

In general,

Violations were dealt with harshly, and police round-ups became common. Surveillance operations indeed led to over 10,000 arrests, 8,500 of which were conducted under presidential warrants, the rest carried out by local justice officials who then reported their actions to the Justice Department in Washington D.C. Most were paroled after a short period of "investigation" into their circumstances, although the arrest itself was a humiliating experience and could lead to loss of employment, social standing, housing or all three.

In California, though, two brothers who failed to register were not interned, even though one of them was 'caught' in a restricted area so there was (perhaps not surprisingly) some inconsistency).

For those who were interned,

Around 10 percent of the 2,300 civilian internees held at Fort Oglethorpe and Fort Douglas were wealthy German-born immigrants suspected of disloyal behaviour, including financing pro-German propaganda. A much bigger contingent was made up of ordinary workers, however, including the destitute and unemployed, and many whose only "crime" was to have been involved in radical politics or labour unrest.... However, even in the case of anti-war agitators, the American authorities were still inclined to investigate each individual case rather than making blanket referrals for internment.