Standard current and discharge current
Standard discharge current is related with nominal/rated battery capacity (for example 2500mAh), and cycle count.

If battery is discharged with higher current - real available capacity will be smaller (it may be much smaller). Discharging battery with lower current will extend real available capacity a little bit.

Currents higher than standard will shorten battery life, lower will extend it (in comparison with nominal cycle count).

Maximum continuous discharge current is a current that will not overheat and destroy battery, but keep in mind that discharging battery with maximum allowed current will reduce battery life significantly and probably real available capacity will be less than nominal.

Here is an example, where you can see how battery capacity changes at diffrent discharge currents



Source: Panasonic LC-R121R3P VRLA battery datasheet

There is no information about maximum discharge current, however it looks like that battery can be discharged with 850mA, but you can't use whole 1.3Ah capacity.

Another example - Sanyo AAA Eneloop NiMh rechargeable battery:





Source: Sanyo AAA Eneloop NiMh HR4U-TGA NiMH battery datasheet