If you like star trek or similar science fiction, you have probably heard of the term, "space-time continuum". Well it is a real thing, as is time, and yes the definition of time still works for all of our day to day scheduling of events and activities.      When things start moving near the speed of light or are in a strong gravitational field, time might seem to go awry. We all have a good appreciation for length, width and height. As fundamental as these three dimensions are to our understanding of the world around us, modern science tells us that these are mixed into time itself.
As crazy as that might sound, Albert Einstein's special theory of relativity has demonstrated that as far as we can measure, space and time are intimately connected and in some cases indistinguishable. The most fundamental definition of time is that it is the variable which distinguishes separate events even if they occur at the same location in space.
Time is a continuous variable like length insofar as it can effectively be split up into ever smaller intervals. Time is what defines the location of a bullet if we want to know where it was at any moment during its trajectory through the air. Time is also something that can be shown to cause conservation laws such as the conservation of energy as well as the conservation linear and angular momentum. A conservation law such as that for energy means that energy can neither be created nor destroyed, it can only be converted from one form to another.
This connection to conservation laws came from a very brilliant female mathematician of the early 20th century, Dr. Emmy Noether. According to Albert Einstein, " Noether was the most significant creative mathematical genius thus far produced since the higher education of women began". Dr. Noether basically proved that for those systems which are symmetric under time reversal, a conservation law has to be in place.
One reason this is so significant is that the only instances where time is not symmetric are when disorder is increased. An explosion or splashing water are examples of where disorder is increased so that a movie of the process played backwards can clearly show which direction is positive time. The second law of thermodynamics in simple language could be stated that all processes on a large scale will increase disorder (sometimes to see this you have to make the scale very large but not always).
Perhaps it is obvious that time is integral to so much if not all of science and technology. Probably far more than many people realize. We know from Einstein's special theory of relativity that when we observe systems moving near the speed of light that their clocks are measured to take longer to tick and that their traversed distances have been shrunk. Their time and distance are intertwined, one effecting and even replacing the other (basically what occurs with electric and magnetic fields).
When the length shrinks in the direction of linear motion, the time grows in exchange. One basically gets traded for the other meaning time and space are intertwined and in this way, they are kind of made out of the same stuff. In nerdy speak this is called the space-time continuum.
From Einstein's general theory of relativity we know that gravitational fields also effect space and time. Our nations global positioning system (GPS) has to make corrections for time running slower on earth than it does in space in order to obtain the highest absolute accuracy for locating spots on the earth. This can be attributed to the acceleration caused by gravitational fields having a similar effect on time and space as the linear motion described in special relativity.
This gravitation-time effect goes to the extreme in black holes where it is believed that time literally stands still. With all this, your watch does correctly measure time as does your cell phone or any other clock. The effect discussed only becomes appreciable either over very long periods of time or at extreme speeds or accelerations. Still time is as real and as interesting as any other dimension, perhaps more so.