Oscilloscope: How are Bandwidth specification and Sampling Rate related?
The bandwidth is related to the analog part of an oscilloscope; the attenuator, amplifier and trigger modules. It specifies the frequency where the signal is attenuated by 3dB.

Simply put the sample rate determines how "fast" the ADC probes the analog signal to gather discrete readings. According to the Nyquist theorem it should be at least twice the maximum signal frequency (bandwidth). If the bandwidth would be only half the sample rate, the resulting reading would then be lower that it actually is. 

On the other hand, frequencies higher than the sample rate (given twice or thrice the bandwidth is available), can not be read without "loosing information" (that's called aliasing). 

See this Keysight document on Scope fundamentals for EE Students. On page 17 you find a typical block diagram of an oscilloscope, page 18 and 19 mention the relationship between bandwidth and sample rate.

From page 17, part of the block diagram (the blueish part is determines the analog bandwidth):



From page 18:


  
  All oscilloscopes exhibit a low-pass frequency response.
  The frequency where an input sine wave is attenuated by 3 dB defines the scope’s bandwidth.
  


From page 19:


  
  Required BW for analog applications: ≥ 3X highest sine wave frequency.
  Required BW for digital applications: ≥ 5X highest digital clock rate.
  More accurate BW determination based on signal edge speeds (refer to “Bandwidth” application note listed at end of presentation)