It's hard to be objective when answering this question as I personally hate this kind of camera work (out of context), but in general, 'swaying' shots and 'shaky-cam' fall under the same catch-all monicker of 'hand-held'.
First used to simulate the hand-held appearance of news reel footage in pseudo-documentaries, the camera form had a resurgence in the early 60s when the new wave of directors felt it their duty to rebel against the established 'old guard' - then it resurfaced after NYPD Blue on TV began shooting episodes with the technique.
The introduction of cheaper, prosumer cameras led to a proliferation of hand-held indie films, which was adopted by the larger studios especially during action scenes in order to give the scene a sense of confusion and urgency.
However, the camera form has been over-used in many cases, and is especially jarring when used out of context - for example, in the recent Man of Steel, the hand-held cinematography was fully justified during the battle scenes and moments of extreme action, but was completely unnecessary (in my opinion) during the intimate conversational moments between Clark and his parents.
This technique often has the effect of pulling the viewer out of the film by drawing attention to the camera move, or in extreme cases, causing nausea.