You can:
Buy a multimeter and test your battery, charging system and a whole lot more with such a tool. They are silly-cheap. Tool Station & Screwfix have decent ones for around £7.00 and are endlessly useful.
Look for a YouTube video on using one to check your charging system/battery. It should be able to maintain around 13-14.6 volts, lights on main beam, heated rear window on, engine idling on a Modern Car.
Get your battery tested at Kwik Fit or Halfords for either free, or a fiver dependent on current offers. But, do NOT buy from either place. The former will in my experience be low-quality, but fairly cheap. Halfords will be of decent quality, will be fitted for £6, but way too expensive. Easily Double+ in comparison to one from a local car accessory shop.
The Bottom-Line Real-World answer, without going into the endless possible yet unlikely scenarios of why you needed to push-start your vehicle is:
When you got home, the next time that you tried to start the car, did it start?
Easily?
If so, it is 95% certain that it's simply the battery. You need a new battery.
Or, was the battery even more flat?
If so, you have a charging system fault. Meaning that about the same odds of around 90% either the alternator is defective or the alternator's drive belt is loose/broken.
Buy a battery mail order from eBay, giving you reg number to cover youself if the wrong part arrives. Exide, Lucas, Bosch, Hella are all decent makes.
45 years experience as a motor mechanic. Time served City & Guilds qualified with HND in Electronics & Electrical Engineering. Own garage business as a Citroen Specialist.
The bottom line is, it will be the battery. If you managed to get home, the altenator's charging. Buy a new battery.
It was 1998 that I last came across a car that did not fall into the two scenarios. Modern cars are in some ways a paragon of dependable machinery.
Good Luck x