In physical terms, dimension refers to the constituent structure of all space and its position in time. Time is same throughout the universe. So, how can time be a dimension?
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The point of considering time a dimension is that it transforms as a dimension -- the math in special relativity works out to be exactly that of a 4-dimensional geometry with a Minkowski metric.
The most well-known among laymen example of this is in the invariance of the spacetime interval -- in Galilean relativity, $dx^2+dy^2+dz^2$ is an invariant under rotations as well as boosts -- individually, $dx$, $dy$ and $dz$ may change when you rotate your axes, but their Pythagorean sum remains invariant.
In special relativity, boosts matter, and $dt^2-dx^2-dy^2-dz^2$ is the invariant quantity under this transformation -- this is a generalisation of the Pythagorean theorem to a 4-dimensional Minkowski geometry.
The most general "proof" that time can be considered a dimension similar to those of space, though, is in the full Lorentz transformation, not just of its norm. The Lorentz transformation between $t$ and $x$ (the axis of motion) can be written as:
$$\begin{gathered} t' = t\cosh \xi - x\sinh \xi \hfill \\ x' = x\cosh \xi - t\sinh \xi \hfill \\ \end{gathered} $$
Which takes almost exactly the form of a rotation transformation (it's actually a skew), where boosting into a velocity of $v=\tanh\xi$ is just a rotation in the t-x plane, analogous to $\tan\theta$ for spatial rotations.
Abhimanyu Pallavi Sudhir
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Dimension comes from Latin dimensionem "a measuring". We can measure time using a clock and therefore it's a dimension.
Larry Harson
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