Are you one of those kinds of people who can't go anywhere without a watch? My husband is; I used to be that way, until I lost my watch once. Then I realized that if you're looking around, you can almost always find a clock somewhere. Of course, this may put you in the situation where you can't find one and you have to ask someone for the time. As I'm not shy, that's not really a problem.
The sense of time changes over the course of one's life. I remember feeling like a class I disliked in elementary school was going to last forever. Now I can guess pretty accurately when five or ten or twenty minutes have gone by. I've even heard that this has been studied scientifically, and it really does change.
It's not only the internal clock that gives you a sense of time, though. It's events. This is where you can start putting on the worldbuilding glasses if you like. I spent so long listening to school bells that I still get an adrenaline rush when I hear the bells at my son's school and feel I might be late. School years versus summers have always divided my life, because my parents work in the university setting. A working life, though, perhaps that's measured differently. Holidays mark time in our lives. Birthdays. Once I was an adult, though, I found that it became harder to remember which gift came from a year ago, and which from three years ago. Gee, I thought, time is running together!
And then I had kids.
Kids change your sense of time like nothing I've ever experienced. When my son was born he used to nurse for 45 minutes every two hours, twenty-four hours a day. For a straight month, there was no day and no night, only this endless sequence of feedings and attempts at sleep. I had to restart everything. Once I had day and night again, I found that minutes would creep by. I'd struggle to get through the last ten minutes before my husband got home, for example. Hours would feel interminable. Yet at the same time, the weeks would fly by. It's that funny feeling where you're so busy you can hardly breathe, but at the end of the day you can't really identify a single thing you did.
The clock-style life is treated differently by different groups (the California party-goers who are always half an hour late, the BART trains which are usually within 2-3 minutes of on time, the Japanese trains which run brutally on time and some of which come every 45 seconds during rush hour, etc.). But it's not the only one. I think about farming families who used to wake with the sun and go to sleep with it, and whose years are measured by temperature, frost and season.
There are also different ways of measuring time. This whole clock thing is convenient, but the clock, and its "clockwise" - turning hands were decided upon by consensus at one point. In ancient Japan they didn't use the same hours we do, but would measure time according to slightly larger blocks named for animals - "the hour of the ox", for example. So when you're designing your world, don't feel you're restricted. Pick a cultural and environmental reason for the way your people measure time.
Wikipedia has a great entry ("Second") which links the current measurement of our seconds to fluctuations in the element cesium. I once decided to create a time measurement system that was influenced by binary calculations, with 64 seconds in a minute and 64 minutes in an hour, all of that measured on the basis of the observable movement of a nearby star.
There are lots of options, but for now I'm out of time. :-)