The way disposable diapers absorb the liquid or urine is due to either SAP or wood pulp fibers.
Cheaper brands of disposable diapers may use less SAP and more wood based pulp as the material that absorbs the urine.
In diapers such as Pampers, Huggies and Luvs they use SAP which is called Super Absorbent Polymer and when urine or any other liquid comes into contact with that SAP gel it sucks it up and swells.
That's why when a kid who's in diapers wets their diaper it begins to swell and the diaper will continue to swell up as the child urinates more and more until it absorbs all it can.
Then at that point the diaper leaks because the SAP has absorbed all it can absorb.
When disposable diapers first came out they didn't really contain SAP but instead contained the wood pulp as do many current adult disposable diapers do today.
Some higher end better quality adult diapers contain SAP but the cheaper ones mostly contain the wood pulp.