How important are citations when applying for jobs or promotions?

In my experience, citation counts are considerably less important than recommendation letters, but they still matter, especially for tenure. For fresh PhDs, high citation counts are definitely helpful, but they're not a hiring requirement for most fields in CS. But for tenure, it's really hard to build a successful case without at least one high-impact (post-thesis) publication.

Smart committees know to gather citation counts, publication counts, acceptance rates, impact factors, h-indices, and other quasi-objective numerical data from reasonable sources and to compare them with peers in the applicant's field. (For my promotion cases, for example, my citation counts were mined from Google Scholar, not ISI, and I was compared against other theoretical computer scientists, not other computational-geometers-who-play-with-surface-graphs.) Alas, not every committee is smart.

To answer the part of the question about freshly-minted doctorates, citation counts are not normally considered as a critical component, either for academia or for industry. Or perhaps I should state that the lack of citations for recently published literature is not a major obstacle, particularly in fields with long "half-lives" for research papers. (Some of my papers only got cited after a year or more following publication.) However, a high citation count for a paper indicates that the student is doing potentially impactful work, and that can carry weight with a professor or manager doing hiring in either academia or industry.