For centuries cooking was more art than science. People did things by taste and feel alone without really considering the science behind it. Initially people studied the chemical and physical process of cooking without considering how they might apply that knowledge to make food more beautiful and better tasting. Molecular gastronomy sprang out of food science but rather than taking a commercial route it was something chefs began to explore seriously. Now this technology is even becoming available to home cooks.
There's a great deal of debate about what exactly defines molecular gastronomy, who is really practicing it and whether or not it is a good thing for chefs to use but there's no arguing that it has changed the way we look at food and cooking.
The major innovations that have come out of the combination of science and cooking work on changing and modifying the physical properties of ingredients. These include spherification, gelification, emulsification and foams. 
Foams were one of the early innovations in the field. The basic idea is to set a liquid with some kind of gelling agent or stabilizer and use a nitrogen-cartridge whipped cream canister to force it through at high pressure and cause it to make a stable, aerated foam. This makes for lighter textures and a silky mouthfeel.
Gelification is a way to create all sorts of fascinating textures and sensations with food. The basic idea is to take a liquid and turn it into a solid form using a gelling agent. There are a wide variety of gelling agents including gelatin, pectin and agar agar. A common way to use gels in molecular cooking is to create a dish where the textures and flavours diverge. For example a chicken soup in the form of gelled cubes with all of the usual ingredients and their flavour profiles present but in a solid form.
Spherification is a fairly recent innovation in molecular gastronomy. This technique creates spheres with a thin skin that hold a liquid inside them. This is done with calcium chloride and sodium alginate. The sodium alginate causes the liquid to set into a gel and the calcium chloride causes the gel to form spheres. The idea behind spherification is to create little bursts of flavour to enhance a dish and make it look beautiful.
Part of what has come to be called molecular gastronomy is the use of specialized equipment to create dishes. Some of the equipment is traditional scientific equipment that has been adapted and some has been created specifically by chefs to fit their needs. Three of the most common devices are the anti-griddle, the immersion circulator and the sous vide machine.
The anti-griddle was inspired by a piece of equipment developed by Chef Grant Achatz. Achatz is considered to be one of the more innovative chefs working in the framework of "molecular gastronomy". The anti-griddle has a flat metal surface that's cooled to -30 F with circulated refrigerant. This means that most liquid and semi-liquid foods will freeze in 30 to 90 seconds. It can be used to create cold items with a crunchy exterior and a creamy interior. 
An immersion circulator is a common piece of lab equipment that keeps water at a precise temperature for an extended length of time. Chefs began to use it because when foods are vacuum sealed and placed in a water bath at a constant low temperature they cook evenly and retain their moisture without a risk of contamination.
The sous vide machine helps in the process of cooking in an immersion circulator. Basically it allows cooks to vacuum-pack ingredients for sous vide cooking in a safe manner. Great care must be taken in the preparation of any food for sous vide cooking to ensure that there is no growth of anaerobic bacteria in the food but aside from those precautions, it is a simple way to cook. 
All of these substances, items, techniques and ways of thinking are broadening the culinary world's repertoire of available ways to produce flavorful, beautiful food and helping chefs to fuse science and creativity in ways that advance the craft of cooking.