Norm of a Bounded Linear Operator

Sorry to be asking so many questions! I'm confused by the definition of the norm of a bounded linear operator,

If $\displaystyle B$ is bounded we define its norm by the equality

$\displaystyle ||B|| := \inf\{ C: ||Bx|| \leq C||x||, \forall x \in X \}$

where $\displaystyle X$ is a vector space. I don't quite understand how the infimum fits into all of this. Any explanations would be excellent. Then, related to that, how is this the case

$\displaystyle ||B|| = \inf\{ C: ||Bx|| \leq C||x||, \forall x \,\, \text{s.t} \,\, ||x||\leq 1 \} = \sup\{||Bx|| : ||x||=1 \},$

and why can I make this assertion? Thank you so much in advance, and happy new year!