Regular ice is made from freezing water. Water freezes at zero degrees Celsius, or +32 degrees Fahrenheit. (Most of Europe and all scientists use Celsius because we're lazy and it's easier, but the US is still stuck on the Fahrenheit temperature scale.)

"Dry ice" is made from frozen carbon dioxide gas. Carbon dioxide is an odd substance because at regular pressure, it is only found in gas and solid forms: the stuff in the air you breathe and dry ice. So when dry ice "melts" it goes directly into the gas form! At really high pressures, carbon dioxide does form a liquid phase. Anyway, dry ice is much, much colder because carbon dioxide freezes at a lower temperature:-79 degrees Celsius or -109 degrees Fahrenheit!

It's very dangerous to handle dry ice because it will freeze your skin and cause frost bite in less than a minute, causing a painful "dry ice burn". Did you know that in Antarctica, the air can sometimes get so cold that the carbon dioxide in the air will freeze, turning into dry ice? This makes it very hard to breathe the air, because without carbon dioxide, our brain doesn't know when to make our body take a breath. (The brain uses the amount of carbon dioxide in our blood as a signal to breathe rather than the amount of oxygen. Go figure!)

Dry ice is made of solid carbon dioxide (CO₂) while ice is made, of course, of solid water.

Almost all substances have a freezing point which depends on their unique physical and chemical characteristics. I'm not sure,though, how well we understand why a particular substance has the exact melting point it does.

Much of the stuff we deal with on a daily basis is solid, meaning that it has a freezing point (transition from liquid to solid) that is higher than "room temperature," or about 20 degrees Celsius. A lot of room temperature liquids, such as water, alcohol, and gasoline have freezing points below 20 C but, perhaps, not too far below 0 C.

Substances that are normally gases at 20 C and one atmosphere pressure, such as CO₂, oxygen, and nitrogen, have much lower freezing points. The temperature of dry ice is around -80 C or 195 K (195 degrees above absolute zero). Oxygen, argon, and nitrogen freeze at around 65 K.
Neon and hydrogen freeze at 10 to 20 K. Normal helium, as it turns out, doesn't ever freeze exactly.

One interesting thing about dry ice is that carbon dioxide doesn't exist as a liquid at room temperature and one atmosphere pressure. So dry ice doesn't melt (hence it is "dry") but sublimates - meaning that the CO₂ goes directly from solid to vapor as it absorbs heat. The CO₂ cylinders that are used to put bubbles in soda are filled with liquid CO₂ but have to be pressurized at around 17 atmospheres.

Lots of things that are gases at room temperature can be frozen into liquids or solids at cold temperatures. Ice is just water frozen into a solid. Even though dry ice is called "ice", it's not.

Dry ice is actually a gas at room temperature - carbon dioxide - frozen into a solid at about -110 degrees. At ice temperatures, carbon dioxide is a gas - and as you probably know is already in the atmosphere and air you breathe. So if you are playing around with dry ice, you want to make sure to be in a large, well-ventilated room so the carbon dioxide level in the room doesn't get to high as the dry ice melts. By the way, -110 degrees is cold enough to hurt you if you touch it without gloves and definitely shouldn't be tasted (which you probably knew).

Something to notice is that dry ice, unlike water, melts directly into a gas without being a liquid. This is called sublimation. There are colder things than dry ice. 70% of all the gas in air is nitrogen. Nitrogen can freeze into a liquid at -320 degrees. Colder than that is liquid helium which has a temperature very close to absolute zero. Thanks for the questions.

Dry ice is carbon dioxide ice, not water ice.

Carbon dioxide freezes at somewhere around -70 C (I think)whereas water freezes at 0 C. Water ice can get that cold, of course, but if you have dry ice, it takes energy (and a lot of it) to take something from a solid to a gaseous phase, so it stays at that temperature until it evaporates (carbon dioxide does not form a liquid at atmospheric pressure, no matter how hard you try).

As a result, the solid stays at its evaporation temperature. Water, by contrast, heats up to its melting point before plateauing for the same reason.