The noydir command corresponds to the Yahoo! Directory and the noodp command correlates to DMOZ.
When a site is listed in the Yahoo! Directory there is a title and description associated with that listing. So, Yahoo! started showing that title and description as the linking text and snippet text in the search engine results page (SERP) whenver that website would show up (most commonly only the homepage). The noydir command overrides this functionality and gives control back to the website owner to control the linking and snippet text in the SERPs via the Title tag and Meta Description tag on their website. By regaining this control you can adjust those fields whenever desired.
The same functionality goes for the noodp command. However, this is directed at Google. Google partnered with ODP to power the Google Directory so if you website was listed in DMOZ then Google would pull that listing information as the linking and snippet text in the Google SERPs. The noodp overrides that functionality and give you control back.
The most common implementation is to do both commands simultaneously as in your question like such:
<meta name=”robots” content=”noydir,noodp” /> (content field order does not matter)
They can also be listed separately as such:
<meta name=”robots” content=”noydir” />
<meta name=”robots” content=”noodp” />
You can also direct the commands to the specific bots as such:
<meta name=”googlebot” content=”noodp” />
<meta name=”slurp” content=”noydir” />