The honeymoon period is a metaphor that is used to compare the new president coming to power and his first few months with the honeymoon of a newly-married couple. Just as the newly-married couple are deeply in love with each other and can't find any fault in their partner, so the country trusts so much in the new president and in the promises that they made in their election campaign that they cut them a lot of slack in their first few months. Interestingly, though, you might want to think about how this isn't true with Presidents, perhaps in particular with Obama.
The Honeymoon for Presidents in this country usually only applies, in a real and practical sense, to their first term in office. We can use President Obama as an example, but really it is true for almost any first term President. When they are elected and take office, they haven't done anything yet. What people remember are the promises of the campaign, and their hope that things can change and improve in their lives and the country under the new administration. While there are no accomplishments yet, there are also no failures, disappointments or liabilities.
For some Presidents, the honeymoon is shorter than others. George W. Bush had a fairly short honeymoon because his election in 2000 was controversial. John F. Kennedy had the failed Bay of Pigs invasion only three months into his Presidency, and it overshadowed and stymied his efforts to do much else in that first year. Ronald Reagan, on the other hand, was able to extend his honeymoon for several months because 1) the Iran Hostage Crisis ended as soon as he took office, and 2) he was a fantastically motivating speaker.
When talking about presidents, the term "honeymoon period" refers to a period of time at the beginning of the president's term in office. This is a time when the president is still quite popular.
The idea here is that each president comes in pretty popular. He won the election and people have high hopes for what he will be able to accomplish. At this point in his presidency, they are willing to give him the benefit of the doubt because they hope that he will govern in ways that they like.
Once the president starts to take controversial actions, the honeymoon ends pretty quickly as people get angry at him no matter what he does.
Typically, it is the grace period that new presidents generate as they become the new head of state. This period is generally free from bad press because of all the new appointments, and with all of the settling in in Washington, DC that has ro be done. New Presidents a are given a lot of slack in their first 100 days.