ConnectionTimeout a property of Connection class in ADO.NET, is the time you would wait, for connecting to a given database, before flagging a connection failure. Default value is 30 seconds.
new SqlConnection().ConnectionTimeout = 10;
CommandTimeout a property of the Command class in ADO.NET, is the time you would wait, for a command (query, stored procedure, etc.) to return result set, before flagging an execution failure. Default value for CommandTimeout too is 30 seconds.
new SqlConnection().CreateCommand().CommandTimeout = 10;
Unlike ConnectionTimeout which are part of connection string, command timeouts are defined separately (hardcoded or as appSettings). Setting both of them to 0 (zero) would result in indefinite wait period which is generally considered a bad practice. You ideally want to set both of them to in accordance to your performance SLAs. Though at times, while running data heavy background jobs / workflows you may want to set your CommandTimeout to a larger value.