- How important is citation count for fresh doctorates? Is this number more important for joining the academia than the industry?
- What is the importance of citations for someone seeking tenure? Does this depend on the field, as papers in an esoteric area can be expected to have few citations?
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.
You already got some nice answers, but let me share my personal experience. I am a fresh PhD and I got hired on aP position after 3 months of graduation. There was another candidate (my friend) who also graduated same time with me, he had 14 times more citations, 6 times more total impact factor, and almost 10 times the number of publications. But he graduated from a university ranked in 400 and I graduated from world top 20. He published most of work by collaborating with different students from developing countries, while my publications were totally focused on my thesis work. He had 100+ publications during PhD but no patent, my all publications were registered for patent and 2 were bought by the industry.
I was convinced that my friend will get this position not me, but the result was surprising I got hired while he was rejected. So, in my opinion sometime citations and number of publications has minimal impact and you may get hired based on from where you are coming and what exactly you did during PhD. Focused, well organised, and quality work with few publications and citations are better than low quality large number of publications with high citations and high total IF.