First order thoughts are straightforward thoughts - 'There's a chair.'. Second order thoughts are thoughts about thoughts - 'I knew that was a chair ...'. Quite high-order thoughts are actually relatively common: 'I can't imagine why you thought I even cared whether he shared my taste in chairs?' - fifth order?
Anyway, an idea which recurs in several different contexts is that consciousness has something to do with higher order stuff: either our awareness of ourselves or thoughts about thoughts. Self-awareness is often considered a special, or even the only true, form of consciousness. Great significance has been attached to experiments which seem to show that while most animals treat their reflection in a mirror as another animal, some of the great apes realise that it is themselves they see (they betray this awareness by reaching up to touch a spot on their faces which they were unaware of until they looked in the mirror). Perhaps the influence of Descartes ('I think, therefore I am' has something to do with the special place which thoughts about ourselves have been given. For sceptics it is also appealing to think that the impression of self-hood might come from turning on ourselves a faculty which developed to help us respond appropriately to other organisms.
The idea that higher order thoughts are special - that conscious thoughts are those we are aware of thinking - remains very appealing. Higher order processes certainly crop up incidentally in many theories about the mind and few would deny that they are at least a feature of consciousness. The view that they are essential to it can be traced back as far as Locke, and comes in many different forms.
One school of thought here is that conscious thoughts are somehow inherently about themselves as well as about their other targets; another version, expounded by David Rosenthal, maintains that thoughts are only conscious when there is a separate higher-order thought (HOT) about them. Neither view sits completely happily with the normal perception that most of our conscious thoughts are just not that complex. Many other issues also remain open - do we need thoughts about thoughts to be conscious, or will perceptions of our thoughts do? Higher-order theories are generally compatible with other theories about the nature of consciousness, and perhaps this is in the end why they haven't made more impact - no-one has succeeded in formulating them in a way which makes them controversial enough to be really interesting.