You can use the Merge Selected Features command to join geometries. This will work best for overlapping geometries, as they will result in a single part. If the parts are discontiguous (even if in a very small way), then they will be considered separate parts and you will end up with a multi-part geometry. There are two modes for this selection: intersection (features where any point is contained in the selection area) and containment (features where all points are contained in the selection area).
To merge selected features into one
- Select the features you want to merge.
Edit > Merge Selected Features
The features you have chosen are merged onto one feature containing each of your selections. The previous geometries will be removed.
Splitting a multi-part feature into separate features
- Select the feature to split
Choose Edit > Split Selected Features
The feature you have chosen will be split into one new feature for each independent part in the existing feature. All feature data will be copied to each newly created feature. The original feature will be removed.
In the case of complex polygons with islands and holes, the islands will be separated into new features and the holes will be assigned to the closest encompassing island.