Science, sometimes called science points, is needed to unlock new parts in the tech tree. It is obtained by performing different scientific activities at different locations and then either returning to Kerbin and recovering the craft or sending your scientific analysis home through an antenna.
Crew reports can be created in every manned command pod. Each command pod can only store one report at a time (Kerbonauts seem to be unable to memorize more than one at a time), but they can be sent home with an antenna without penalty allowing the crew to create a new one.
Experiments are parts like the Mystery Goo Containment Unit and behave similar to crew reports. They can be performed in different flight phases and on different celestial bodies returning different results. The biome, however, does not affect them unless the vessel has landed. Just like crew reports, each experiment part can only store one result at a time, but a result can be sent home with an antenna to allow the experiment to be performed again under the same or under new conditions. There is a penalty for sending it home instead of recovering the performed experiment, but when the craft has a steady supply of electricity to supply its antenna this doesn't matter much because an experiment can be repeated indefinitely resulting in the same science output as would be possible through recovery.
EVA reports are done by kerbonauts on an EVA. Like crew reports they give different results depending on altitude and biome if performed low enough.
In contrary to crew reports and experiments, a kerbonaut can memorize a different report for each condition he performend an EVA in during the mission.
Surface samples can be taken by kerbonauts when performing an EVA on the ground (or in the oceans) of a celestial bodies. The results vary by biome.
Recovering a craft after it landed on Kerbin also yields science points by itself. Its value depends on how far the craft flew. Recovering a craft after a suborbital flight returns less valuable (but different) results than one which orbited the Mun, for example.
Performing the same activity multiple times in the same environment yields additional science points, but with diminishing returns each time the activity is repeated.
When an experiment is recovered after landing on Kerbin, its scientific value is reduced by 80% of its current value when it is performed again under the same conditions.
When it is transmitted instead, this reduction is divided by the transmission efficiency. So an experiment with a value of 10 and a transmit value of 60% will lose 10 × 0.8 × 0.6 = 4.8 science value when transmitted, which means it will yield 10 - 4.8 = 5.2 science the next time it is performed.
As a result it doesn't matter if the player transmits lots of times or returns a few times. In the end the total scientific yield of each experiment will always approach 125% of its initial value.