In a more qualitative sense, if you have a chemical reaction, what's happening on a molecular level is you generally have a transfer of electrons. Depending on the reactants and a few other variables (this is very non-specific and simplified), you can have a reaction that absorbs heat from the surroundings (endothermic) or gives off heat to the surroundings (exothermic).
As for your question about pressure, you have to keep a few things in mind in addition to thermo. From only thermo, you find that pressure and temperature are related--for a constant volume, if temperature goes up, so does pressure. So, in that respect, if you have an exothermic reaction that gives off heat, it will accordingly raise the pressure. This kind of makes sense when you think about it--it's why you don't heat closed containers (e.g., throw hair-spray bottles on a campfire...or maybe you do, but you know the result from the rapidly increased pressure).
Now, the other factor contributing to an increase in pressure is the release of gas. Perhaps more than heat alone, the release of a gas can increase the pressure incredibly in a system. Ever see the baking soda volcano when you were a kid? If you add vinegar to baking soda, what happens? You get a lot of CO2 (carbon dioxide). If one of the products of your reaction is a gas, you will get an increase in pressure much faster than just an exotherm.
BTW, this all assumes you have a closed system. If you have an exothermic reaction that gives off gas, but it's open to the atmosphere (like in an open beaker), you get no pressure increase (but I assumed you'd have figured that out).