Great Britain vs the Continent
Whenever something is referred to as "continental", you can be sure it's meant as the opposite of "British". The popular myth of the newspaper headline reading "Fog Blankets the Channel; Continent Cut Off" is another example of the attitude that spawned this dichotomy.
In this case, a continental breakfast is the opposite of a full English breakfast and that is where the term originated.
Traditional English Breakfast
On the British Isles, a full breakfast (called a full English breakfast, full Scottish breakfast, full Welsh breakfast, or full Irish breakfast depending on the country; I'll use English in my answer since that's the most common term), is a warm — mostly fried — breakfast. Its exact constituents vary, but it usually consists of scrambled eggs, sausages and bacon, baked beans, fried tomatoes, and fried potatoes, sometimes with black and white pudding, or with fried up leftover vegetables from last night. It's quite a heavy meal.
In the rest of Europe, breakfast is usually mainly bread with cheese, slices of meat, or sweet or savoury spreads. Breakfast cereals, sometimes. The only ingredients that are usually eaten warm are eggs or porridge. It is a bit lighter meal than an English breakfast
Hot beverages such as coffee or tea are common to both.
The main difference between a continental breakfast and a full (English) breakfast is cold vs. hot.
A continental breakfast can be served cold, eggs having been boiled beforehand. The only hot ingredients might be the beverages — coffee, tea, hot chocolate. This reduces cost for the hotel and allows for the breakfast to be served in just about any room; it's not unusual to see the hotel bar being used for this if the hotel serves only a continental breakfast.
The hotel doesn't need any cooks to be present, just waiting staff to serve it.
If a hotel serves both a continental breakfast and a full English breakfast, the latter may be offered at a premium, since the costs to the hotel are higher as well.
If you've had a continental breakfast that included scrambled eggs and other hot dishes, it most likely was a breakfast buffet, where the hotel offers something extra to its guests, ending up a bit in the direction of a full English.