The battery in your car is 'lead acid' and that type of battery used to be common in golf trolleys. Just as some cars (e.g. diesels) require heavy duty batteries, you'd need a heavy duty battery for your golf trolley if you wanted to be able to do a couple of rounds without recharging it. So you need to look at at 'Amp-hour' (Ah) rating of any battery you're considering buying. A bit of googling suggests that a 16Ah battery might be offered as an '18 hole' one, whereas a 24Ah battery might be sold as '36 holes'. (If your old battery was, for most of its life, up to to your needs, simply look for one with the same rating as is on its label but remember to check its measurements too!).
The main advantage of lead acid batteries is (relatively) low price; their big disadvantage is weight. A lithium battery weighs only about one quarter of what a lead acid battery does but costs far more.
Unlike car batteries though (which are constantly charged whenever the car is driven), batteries for golf trolleys need to be capable of 'deep cycling'. (That means that they can be frequently discharged and then recharged). While some types of lead acid battery can cope with deep cycling, other types are more suited to it. As well as (expensive) lithium batteries, there are both absorbed glass mat (AGM) and gel cell batteries; there's little difference between them except gel cell batteries might have a slightly longer life.
So a gel cell (or AGM) battery might be best for you - but do check both the 'Ah' rating and its physical size.