The most basic trade-off in the human experience is what you do with your time. In any given period, you can focus mainly on only one given task. The limitation of physics that you can only be in one place at any given time (above the subatomic level) places a restriction on what you may be able to do with your time.
You can often trade-off time for money. For example, you can hire someone to mow your lawn for a given sum and in exchange you retain the time that the job would have taken. Now you can choose to use your time editing on Wikipedia! Or, you can choose to do another task that will earn you more money, or is more pleasurable to you, than mowing the lawn. This gives rise to the common idiom that time is money. The downside of this trade-off is that you don't have your money anymore.
A classic trade-off in business is the trio of time, money and quality. It is generally considered that only two of the three can be anchored at any given moment. Given enough money and attention to quality, one can get man to the moon and back by 1969. Microsoft can put out a new operating system every couple of years if not too much attention is paid to quality, or even quicker for Windows ME.
These trade-offs are ubiquitous in our idiomatic expressions. A stitch in time saves nine. Give me quality or give it to me now.
In engineering, trade-offs abound. It is rare for a bridge or a building to be functional, beautiful, built on time and within budget. Examples of meeting all of these goals are so rare that when they are achieved, it is considered a modern marvel. The Golden Gate Bridge is a prime rare example where few engineering and aesthetic trade-offs had to be made.
In computer science trade-offs are viewed as a tool of the trade. A program can often run faster if it uses more memory. It can be developed faster if it doesn't run as fast. It can be optimized for space or speed, but at the cost of longer and more complex development cycles.
Governmental trade-offs are among the most controversial political and social difficulties of any time. All of politics can be viewed as a series of trade-offs based upon which core values are most core to the most people or politicians. Do you feed the poor or send men to the moon? Do you try to do both and create a deficit that leads to an economic recession? When is the cost of war (socially and economically) justified over the cost of peace?
There are few areas of life where trade-offs don't have a significant and serious impact.