Democrats vs. Whigs: Democrats believed that government should leave business alone, neither helping nor hindering it. This philosophy is known as laissez faire. This way the people would not be burdened with taxes to support big business, and everybody who wasn't too lazy or too ignorant would be able to take care of himself.
The Whigs believed government should protect industry with tariffs on imports, with grants of monopolies, with construction of harbors and railroads, with a national banking system. This had appeal to northern industrialists and farmers who needed railroads or other help getting their produce to market. This did not have appeal to most common people because they did not want to pay more taxes to support big business and they did not want to pay higher prices for what they bought because of the tariff. Southern planters supported the Whig party because they believed it provided better law and order. They thought too many of the common people of the north who supported the Democrats were rabble who endangered law and order.
Here is a video about this subject:
The Whig party (1834) formed in opposition to President Andrew Jackson and the Democratic party. It only lasted for a little over twenty years. There were two Whig presidents who were elected, Harrison and Taylor.
The Whig party wanted to make America stronger by building roads, canals, etc. They viewed President Jackson as being too much like a king, and even gave him the nickname King Andrew I. Democrats also believed in agriculture while Whigs believed in industry. The Whigs also favored a federal government while Democrats favored state government.
Slavery is what ultimately split up the Whig party because most northern Whigs believed in abolishing slavery, while most southern Whigs thought the opposite. This is when the Republican party was formed.
When the Whig Party was formed in 1834, there really was no ideological or policy-based difference between their party and the Democratic Party. Instead, the Whigs were really just a loose coalition of people who were mad at Andrew Jackson. You can see this, for example, in the fact that 28 of the 41 Democrats who voted against the recharter of the Bank of the US all switched and became Whigs.
As time went by, Whigs tended to be more conservative than Democrats. The Democrats were more likely to embrace immigrants and states rights. The Whigs were more nativist, Protestant, and in favor of a strong national government.
These two words refer to members and supporters of different political parties. Democrats are members of Democratic party, and Whigs are member of Political Party that may be popular party in some countries but has other formal name which may vary from country to country. The term Whig first developed in England, where the comparable name for opponents of Whigs is Tories.
However as the question appears to be from student in US, I will restrict my answers as applicable to the US.
In United States Whigs refers to the members of the Republican Party. AS a matter of fact this party, formed in 1834 was originally called American Whig Party, and adopted the name of Republican Party in 1854. It is one of the two major political parties in the USA, the other one being Democratic Party, which is older of the two parties.
Over the years Policies of both these political parties have evolved and changed. Though both these parties have their strong supporters, and both the parties take different stands on specific issues emerging from time to time, as things stand today there is not much of basic difference between the political ideology of the two parties.
The Internet link referred below gives compares the alternative stands of Democratic and Republican parties on a number of issues as in 1999.
The second link compares views of Barack Obama (Democrat) versus John McCain Republican before presidential election in 2008.