For big planes, typically anything much larger than mid-size business jet right through to 747/A380 size, there will exist a fairly healthy margin between the engine thrust/power needed to get airborne and climb out and a healthy rate, and what the engines can deliver with the power levers wide open (for most conditions!). In those circumstances it doesn't make a great deal of sense to thrash the engines, and can indeed make life safer all around if you use just enough engine-ooomph to get up and away.
For those scenarios there is a fairly well trodden path, and it goes something like this...
As a previous poster alludes to, quite often an aircraft will happily take-off from a runway for a large range of ambient temperatures. However when the ambient temperature increases, a number of things happen: the engines typically generate less thrust, the density altitude increases, the IAS/TAS split becomes larger, and so on.
When a large aircraft manufacturer certifies an aircraft, they will conduct a large number of take-off and climbs for a large range of conditions. They will then further in the background take that data, identify trends, and sensibly extrapolate out until you now have a data-set which can tell you exactly how an aircraft will behave for any given runway, OAT, wind, slope etc.
So, back to the day where we're sat in the cockpit, we don't have many passengers, the aircraft is light, and we have a really long runway to take-off from. How much thrust/power is just enough? Well we can play a little thought experiment here: If I used the runway I'm on today, what in theory is the hottest ambient temperature I can successfully take-off with? Fortunately my voluminous data set from the aircraft manufacturer can go through all of those factors and arrive at answer: ASI, you could take-off today up to 45 degrees celsius, say.
What I can then do is consult my engine data and ask; at 45 degC, what engine setting am I going to get? Typically for jet engines this is measured by engine fan-speed (N1), or engine pressure ratio (EPR). I might find out that on a 45 degC day with the throttles wide open, the engine will give me 91% N1.
But it's actually really cold out today, say only 10 degC at the airport, what good does that nugget of knowledge do for me? Well what it means is I can push my throttle forward to get the 91% N1, and the engines will give me just enough thrust to get off the runway and climb away. I have in effect lied to myself about the temperature, and set the engines to a lesser thrust than they could otherwise achieve with full power in today's colder weather.
This approach all in all is called a FLEX derate, and is relatively common amongst Boeing and Airbus (barring differences in terminology). The success of this approach is based on the following things: A really expansive and well validated set of data from the manufacturer, some very accurate engine gauging and predictable engine behaviour, and a rigorous set of SOPs to make sure that the soft and squidgy chaps up front input the right numbers, get the right numbers out, and apply them correctly.
Coming back to the average light aircraft...well, you're basically out of luck on the first two counts straight away...