There are some great answers and info here, but I am going to add my nickle's worth anyway! 
First thing you need to know is that the terms turtle, tortoise, and terrapin have both a specific scientific meaning, and a common meaning. To make it even more fun, the common meaning changes by where you are in the world.
Scientifically-
All shelled reptiles are turtles. While many people prefer the term 'chelonian', the more common term, even in research papers, is turtle.
A tortoise is a turtle from the Family Testudinidae. The true tortoises do not need to enter the water to hunt, eat, mate, hide, feel safe, etc. but more importantly, they have specific anatomical issues- most notably the heavy domed shell (except pancakes) and 'elephantine' legs- a better way to say this may be to say that they have stumpy legs with toenails but no visible toes.
'Terrapin' does not have quite as 'locked in' a scientific definition as the others but is most commonly used for turtles that inhabit fresh or brackish water.
Regionally-
In the US, we use tortoise for the true tortoises, terrapin for the diamondback terrapin, and turtle for everything overall. In some Southern parts of the US the term terrapin is also used for turtles used for food.
In Europe, tortoises are land-dwelling turtles (including the American box turtles); terrapins are the semi-aquatic, pond, or basking turtles; and turtles are the overall term and the term used for marine turtles.
In Australia, they are not real consistent, but often use the term tortoise for native fresh water turtles.
Within turtle groups, you sometimes hear turtles divided up a little differently-
- Chelonians or turtles for all of them
- Tortoises for the true tortoises
- Terrestrial turtles for the species that spend a good amount of time on land
- Pond or basking turtles for mostly aquatic turtles that are nonetheless often seen basking or near water
- Turtles for fresh water turtles that are rarely seen out of the water
- Terrapins for brackish water species
- Sea or marine turtles for ocean-dwelling species
Box turtles-
These guys are interesting in many ways. Physically, they are certainly not true tortoises- distinctive toes, thinner shells, different skull structures, etc. Anatomically they are obviously more related to sliders and other basking turtles than to any true tortoise. However, they really don't need water for feeding, mating, etc. (Cooling off in water or mud does not really count here.)
However, they also have terrapin in their names in the genus Terrapene... which is kind of a misnomer. The word terrapin is from Native American words for turtle- torope, turepe, or tulpe and came to us from the French. As far as I can tell, it is used for the box turtles in recognition of the regional name for them, not because they are considered terrapins by most standards.
First thing you need to know is that the terms turtle, tortoise, and terrapin have both a specific scientific meaning, and a common meaning. To make it even more fun, the common meaning changes by where you are in the world.
Scientifically-
All shelled reptiles are turtles. While many people prefer the term 'chelonian', the more common term, even in research papers, is turtle.
A tortoise is a turtle from the Family Testudinidae. The true tortoises do not need to enter the water to hunt, eat, mate, hide, feel safe, etc. but more importantly, they have specific anatomical issues- most notably the heavy domed shell (except pancakes) and 'elephantine' legs- a better way to say this may be to say that they have stumpy legs with toenails but no visible toes.
'Terrapin' does not have quite as 'locked in' a scientific definition as the others but is most commonly used for turtles that inhabit fresh or brackish water.
Regionally-
In the US, we use tortoise for the true tortoises, terrapin for the diamondback terrapin, and turtle for everything overall. In some Southern parts of the US the term terrapin is also used for turtles used for food.
In Europe, tortoises are land-dwelling turtles (including the American box turtles); terrapins are the semi-aquatic, pond, or basking turtles; and turtles are the overall term and the term used for marine turtles.
In Australia, they are not real consistent, but often use the term tortoise for native fresh water turtles.
Within turtle groups, you sometimes hear turtles divided up a little differently-
- Chelonians or turtles for all of them
- Tortoises for the true tortoises
- Terrestrial turtles for the species that spend a good amount of time on land
- Pond or basking turtles for mostly aquatic turtles that are nonetheless often seen basking or near water
- Turtles for fresh water turtles that are rarely seen out of the water
- Terrapins for brackish water species
- Sea or marine turtles for ocean-dwelling species
Box turtles-
These guys are interesting in many ways. Physically, they are certainly not true tortoises- distinctive toes, thinner shells, different skull structures, etc. Anatomically they are obviously more related to sliders and other basking turtles than to any true tortoise. However, they really don't need water for feeding, mating, etc. (Cooling off in water or mud does not really count here.)
However, they also have terrapin in their names in the genus Terrapene... which is kind of a misnomer. The word terrapin is from Native American words for turtle- torope, turepe, or tulpe and came to us from the French. As far as I can tell, it is used for the box turtles in recognition of the regional name for them, not because they are considered terrapins by most standards.
