What is sample rate of DSO and what if we have a DSO with low value for sample rate?
What makes those different from mine


They have a faster clock, a faster ADC, and faster processing to store the samples at a higher rate.


  when my scope will get stuck due to low sample rate.


From the Nyquist theorem, to reconstruct a signal with bandwidth \$\Delta f\$ in Hz requires sampling at \$2\Delta f\$ samples per second or higher. 

So your scope won't be able to reconstruct signals at more than 24 MHz. 

Practically, it probably has an anti-aliasing filter to limit the bandwidth to slightly below that (10 MHz? 16 MHz? 20 MHz?) and avoid issues that come when you try to reconstruct a signal just at the Nyquist limit.