Are there amide versions of lactides and lactones?
Amide analogue of lactons are lactams, their tautomeric forms are lactims. Citing from the IUPAC Nomenclature of Organic Chemistry (Preferred names 2013):


  P-66.1.5.1 Lactams and lactims
  
  Intramolecular amides of amino carboxylic acids, $\ce{-CO-NH\bond{-}}$, are
  called ‘lactams’ and their tautomers, $\ce{-C(OH)=N\bond{-}}$, are ‘lactims’. Lactams named in two ways:
  
  &nbsp;&nbsp;(1) as heterocyclic pseudoketones;
  
  &nbsp;&nbsp;(2) by substituting ‘lactam’ for the ‘ic acid’ ending of the systematic ‘oic acid’ name for the parent acid without the amino substituent, and inserting a locant designating the position of the amino group between the ‘o’ and the ‘lactam’. Method ‘lactams’. Lactims are named in the same way, using ‘lactim’ in place of ‘lactam’.
  
  (1) generates preferred IUPAC names.
  
  Examples:
  
  
  
  pyrrolidin-2-one (PIN)
  butano-4-lactam
  
  (…)
  
  
  
  3,4,5,6-tetrahydropyridin-2-ol (PIN)
  pentano-5-lactim


(Note that the numberings depicted are for the preferred names (PINs) based on nitrogen heterocyclics)

There's no (at least IUPAC) term for lactides analogue (‘lactide’ is not used in IUPAC names themselves anyway), but they exist.

In traditional or general names, the Greek letter numbering is used, e.g. ε-caprolactam, or β-lactam four-membered ring part in bicyclic penicillin skeleton.