What files are necessary to upload a shapefile?

A Shapefile is comprised of a set of files that correspond to the geometries, index information, projection information and tabular data. The minimum files necessary are these: 

.shp - the shape format, holding the geometry data
.shx - the shape index format, an index of the geometries for traversing through them
.dbf - the attributes or tabular data for each shape

The .dbf should be encoded in UTF-8 to guarantee a successful import to AmigoCloud. If the .dbf file has a different encoding, you must include a .cpg file (Code Page) containing the encoding in a single line. It is preferred that field names do not contain special characters. Recent desktop GIS software use UTF-8 encoding as well, so most of the time you do not have to worry about this.

More information on the format of a Shapefile look at the article on wikipedia.