In relativity, and all relativistic theories like the standard model and string theory, time is one of the dimensions, but treated a little differently from the space dimensions (a change in the sign of some numbers). This is also true in general relativity, though there we find that the time dimension, as well as the space dimensions, can bend.
In non-relativistic quantum theory, which is still used, space position is a mathematical "operator" which chnges the value of a wave function, and corresponds to the physical operation of measuring a position. But time is not an operator for technical reasons, and reamains just a numerical value as in ordinary life - so many nanoseconds or whatever.
In classical physica, like ordinary physics 101, time is a number again, like we usually have it.
Having different time definitions seems confusing at first, and you wonder how unified all these different theories could be, with so many different "times". But you really have only three. All of relativistic quantum theory is consistent, and the older quantum theory is kept around because it works well at low speeds, for example in chemistry. Likewise the General Relativity theory works very well at astronomical distances, but even the GR theorists believe it will eventually be replaced with something that is valid at really short distances too.