There are many answers to this question, because this is an essential research topic of any dual process model (of which there are quit a lot). Most of them posit in some way that whether people rely on "System 1" or "System 2" (or whatever they are called in the respective flavor of the model) is some combination of motivation (Is person X motivated to think hard in situation Y?) and ability (Can Person X think hard in situation Y?).
To add to the other great answers, a few more examples:
From an interindividual differences perspective, there is the concept of Need for Cognition and an accompanying scale that measures "the tendency for an individual to engage in and enjoy thinking" (Cacioppo & Petty, 1982, p. 116). People who are low in need for cognition are, for example, more likely to process persuasive messages in a heuristic way, also are more likely to fall prey to (at least some) cognitive biases such as halo effects or anchoring, and they stereotype more strongly (Petty et al., 2009).
Also stemming from persuasion research, there is the concept of involvement. In a nutshell, if people don't care so much about a topic (low motivation to think hard), they will be swayed more easily by heuristic cues (Petty & Cacioppo, 1984).
There is also the concept of Ego Depletion, which holds self-regulatory capacity ("willpower") is a limited resource (Baumeister, Bratslavsky, Muraven, & Tice, 1998). According to this model, hard thinking requires self-control. If your self-regulatory capacity has been depleted (because of other difficult tasks, such as suppressing emotions or resisting temptations), you will be less likely to perform well in such tasks.
More recent research suggests that people have different beliefs about willpower and whether the latter phenomenon will occur: Some people more strongly belief that willpower can be drained. It's those people who will suffer more from taxing tasks (Job, Dweck, & Walton, 2010).
Finally (for this answer), there is a host of situational and dispositional variables that will affect whether people are able to engage in hard thinking. For example, people with low working memory capacity may be more susceptible to impulsive influences on their behavior. The same occurs for people who are put under cognitive load, are intoxicated, or under time pressure (for a review, see Hofmann & Strack, 2009).
Baumeister, R. F., Bratslavsky, E., Muraven, M., & Tice, D. M. (1998). Ego depletion: Is the active self a limited resource? Journal of Personality and Social Psychology, 74, 1252–1265. doi:10.1037/0022-35220.127.116.112
Cacioppo, J. T., & Petty, R. E. (1982). The need for cognition. Journal of Personality and Social Psychology, 42, 116–131.
Hofmann, W., Friese, M., & Strack, F. (2009). Impulse and self-control from a dual-systems perspective. Perspectives on Psychological Science, 4, 162–176. doi:10.1111/j.1745-6924.2009.01116.x
Job, V., Dweck, C. S., & Walton, G. M. (2010). Ego depletion-Is it all in your head? Implicit theories about willpower affect self-regulation. Psychological Science. doi:10.1177/0956797610384745
Petty, R.E., Briñol, P., Loersch, C.; McCaslin, M.J. (2009). "Chapter 21. The Need for Cognition". In Leary, Mark R. & Hoyle, Rick H. Handbook of Individual Differences in Social behavior (pp. 318–329). New York: Guildford.
Petty, R. E., & Cacioppo, J. T. (1984). The effects of involvement on responses to argument quantity and quality: Central and peripheral routes to persuasion. Journal of Personality and Social Psychology, 46, 69–81. doi:10.1037/0022-3518.104.22.168