Our understanding of gamma-ray bursts (GRBs) and their afterglows has progressed dramatically over the last few years thanks to the Swift mission and progress in prompt follow-up observations. These advances have led to frequent near-complete coverage at optical and X-ray wavelengths starting shortly after the bursts and often lasting for weeks. This dissertation explores the properties of X-ray afterglows in the context of the canonical X-ray afterglow model, with particular emphasis on the lack of expected jet collimation signature that was commonly seen in pre-Swift optical observations. I find that with slight modifications to the canonical X-ray afterglow model in the context of the Fireball model, by testing many permutations of the closure relations, I can find many more jet breaks than are initially obvious from the data; the remaining residual deficit can be largely resolved by accounting for observing biases. The properties of jet collimation measured from GRB afterglows have direct implications for the inferred energetics of the bursts, which has important theoretical implications. I study the extraordinary naked-eye GRB 080319B as an example of a burst with unusual observational properties of the prompt emission and afterglow suggestive of more complex two-component jet structure than the canonical form. Finally, I explore the expectations for GRB afterglow observations if all GRBs have this two-component structure but are commonly seen off-axis. The implications of this non-canonical two-component jet structure for the total energetics, frequency, and physics of GRBs and their afterglows are far-reaching.
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