The National Hurricane Center is the official hurricane forecast center for the Atlantic basin. If you visit one site, their site is the place to go to get the latest official information. When a tropical feature has reached depression status, the front page will feature what you need to look at. Take a look at the "Public Advisory" which will appear on the front page under the heading of the tropical feature for easy to understand language. The site also features the forecast track for the storm and other products.
This is an excellent resource. It provides wonderful visible satellite images of current cyclones and developing disturbances. It has an incredible amount of other data. This is among the first sites you will want to visit to see if a storm is developing. Under the "Atlantic" heading in the left column, you will see something like "90L.INVEST," which is an area of disturbed weather that is being monitored. This is not something the NHC would be issuing advisories on, though you may see the area talked about in the "Atlantic Tropical Weather Outlook" on the NHC's site. The NRL site, often referenced as the "Navy site," will usually have these areas of investigation listed before any other site. They do not always develop. When a storm reaches depression status, then advisories will be written on the NHC's site. These areas of disturbed weather are numbered 90 through 99. When the number 99 is reached, we go back to 90.
The official names above are selected by an intl. committee of the WMO for tropical & subtropical cyclones in the North Atlantic basin. Other types of storms, such as winter storms, do not have official names.