Basically, the dissolve
tool does what you need. It dissolves all features on a layer into a single one with multiple parts. If your features had a common ID indicating, which features touch each other, and which don't, you could hand this ID to the dissolve-GUI and only features with an identical ID would be dissolved into larger, multipart features. Be aware that dissolving also removes individual attributes.
Since you don't have that ID, you then need to run multi- to single-part
in order to undissolve/split your multi-part geometry into single features. Now your touching polygons will have become one, while the other polygons are still separate features.
Merge
, then runmulti- to single-part
.