These are known as phishing attacks and come in these varieties:
Phishing
An official looking email which looks like it comes from, say, your bank. It’s really a scam to get your account passwords or credit card details.
Spear phishing
This is a targeted version of phishing directed at a single individual rather than a mass email list. Carefully crafted after research on social media sites, they target high profile people in key roles.
Vishing
Voice based phishing using the phone rather than email.
Smishing
As above but using SMS messaging on a mobile phone rather than email.