(Note that larger capacity (more Wh) batteries only mean longer range, not more power or speed!)
e-bike Lithium-Ion batteries come in different sizes. Their size or capacity, and so how far they can take you, is expressed in Watt Hours (Wh), arrived at by multiplying the most usual 36 volt electric motor x the number of Amp hours (Ah) the battery holds. So for an easy example a 36 volt system with a 10Ah battery makes for a 360Wh battery – ie: 36(v)x10(Ah) = 360Wh.
250Wh, 300w, 400Wh, 500Wh, 625/30Wh and lately even 750Wh batteries are the most common sizes, and they are increasingly becoming integrated into the frame to a greater or lesser extent, including discrete internally housed (but usually still removeable) ‘tube’ type batteries.
You consume around 5 – 20 of the battery’s Watt hours for every mile you are cycling with the power on This depends greatly on factors like your own weight, the weight you are carrying, how well maintained your bike is (especially the tyres being properly inflated), hills, headwinds and speed. All these will all make a significant difference to the level of assistance you choose to select and therefore the amount of battery power you use and so how far it will take you.
So, a battery will provide you with power for around 20 to 50+ miles, depending on its size, the assistance level chosen and how willing you are to cycle with the power off or on a low setting for the easier parts of the journey, thus eking out the battery.
(You choosing to pedal vigorously at energy hungry moments like accelerating from rest or hill climbing will help increase the range nicely!)
Bosch have a helpful ‘Range assistant’ here but it’s a bit optimistic for Sheffield’s hillier routes! If you are a heavy rider and/or loaded up you may achieve more like 75% of these predicted mileages.
Battery warranties/eventual life expectancies are based on the overall number of charging cycles performed. Bosch batteries say the cells they will use will still have at least 60% capacity after two years or 500 whole charge cycles (depending on which happens first) and could give usable service for 10 years or more.
Lithium batteries, particularly cheaper ones, can pack up after a year or two at worst if misused (particularly being left uncharged). The more expensive mainstream ones on all good quality e-bikes should last up to 10 years or more if used fairly frequently and kept reasonably well charged, but all will gradually lose their capacity and therefore range as they age.
We are 7 yrs in with one of our regularly used Bosch batteries, and it is still going strong.
Good battery overview from Cycling UK here and from also from electricbikereport