Order refers to the order of the polynomial that is effectively used in the upsampling process. A zero-order hold will hold the previous value as a constant to insert all new interpolated samples, until the next sample, while a first-order interpolation will insert samples linearly between the existing samples, and so on.

Often interpolation is done by upsampling by inserting zeros between the existing samples to have samples at the higher rate, and this is followed by an interpolation filter which "grows" the zeros to the ideal (or close to ideal as limited by the filter design complexity) values for the interpolated signal.

To see how this works, and the additional distortion that a zero-order hold approach would introduce (known as passband droop) please consider how a zero-order hold mathematically can be viewed as the convolution of our original signal with a (sampled) pulse as depicted in the figure below:
   
[![zero-order hold convolve with pulse][1]][1]

Convolution in the time domain is multiplication in the frequency domain. The Fourier Transform of a pulse is a Sinc function (and a sampled pulse is the similar Dirichlet Kernel which is an aliased Sinc). This explains the "pass-band droop" distortion that a zero-hold upsampling would introduce as shown in the magnitude of the frequency response for the sampled pulse in the figure below: 

[![Multiply with a Sinc][2]][2]


  [1]: https://i.sstatic.net/aQ3ZG.png
  [2]: https://i.sstatic.net/CSF3M.png