A car battery's main purpose is to start the engine, once the engine is running the alternator supplies power to the car by keeping the battery charged, therefore it's the battery's ability to start the car that's the important issue, once it's started capacity isn't really a major factor. The 2 most important statistics for batteries are capacity measured in Ah (Ampere hours) and Cold Cranking Amps (CCA).
Ampere hours (Ah): this is a measure of the capacity of the battery, i.e. how much power it can deliver when fully charged, and for how long. A 60Ah battery can deliver 6 amps for 10 hours or 12 amps for 6 hours.
CCA: Lead acid batteries perform much worse in cold temperatures, the statistic I have read is batteries lose 30-40% of their power at 32°F (0°C) and by 0°F (-18°C) they lose 60%. CCA stands for Cold Cranking Amps, it's an industry standard of how many amps a battery can produce at 0°F. CCAs vary on batteries due to the materials and design of the batteries, 2 batteries with the same Ah can produce very different CCA.
You want to buy a battery with the capacity (Ah) required to deliver the power your car needs to run its alarm system when parked, power the computers and fuel pump(s), and then crank the engine to start. If you buy a battery with less capacity then required you will have trouble starting and the battery will run out of charge faster when parked. If you buy a battery with more than you need your battery will last longer when parked and have more power to start the car. If you go too big your alternator may not be able to charge it. If you go very small your alternator may overcharge the battery and damage it, although chances are a battery that small wouldn't start your car anyway. It makes no sense to skimp and put a lower capacity battery in your car, it's probably going to leave you stranded somewhere.
Using a battery with a higher CCA than your car's spec hurts nothing but your wallet unless you live in extreme cold. When I lived in Wisconsin I uprated my battery for more CCAs and it started in -30°F, so it was money well spent. Where I live now rarely gets below freezing so I wouldn't spend that money. Getting lower CCA than your original battery means that in cold temperatures your car may not start, but if you live in a mild climate you'll probably get away with it.