What are the key core issues between parties on the right and parties on the left?What are the key core issues between parties on the right and parties on the left?
This answer will differ to some extent depending on what country you are asking about. In general, however, the core issue that separates parties on the right from those on the left is their attitude towards government involvement in the economy.
Parties of the right tend to believe in unfettered capitalism. They, like the Tories in Margaret Thatcher's time, believe in private enterprise rather than in government ownership of key industries. They, like Republicans in the US today, believe in fewer government regulations regarding such things as the environment. In other words, they want businesses to be able to do more or less as they please.
By contrast, parties of the left believe more in government intervention. They, like the Democrats in the US today, believe in things like having the government require that people buy health care. They believe in having the government regulate (as many parties in Europe want) financial markets. These are parties that feel that capitalism needs to be restrained by the government.
In general, parties on the left tend to want to even out the economic equalities between rich and poor by taxing the top earners more heavily and using the money to provide social services and education to those in the lower income brackets. While not universally so, parties on the left tend to favor a woman's right to choose an abortion and gay marriage (notable exception: socially conservative Democrats).
Parties on the right want a favorable business climate and low taxes, though in recent years they have not generally adopted policies of low spending. That was more true in the days of Barry Goldwater conservatism. The right also often favors larger defense spending and enhanced law enforcement powers, while often opposing abortion and gay rights (notable exception: Log Cabin Republicans).
I have always looked at right wing and left wing as being more or less accepting of societal rights.
For example, a right-wing follower is most likely to support staunch religious and nationalistic ideologies. This idea is supported by the fact that The Department of Home;land security has tagged some extreme right-wing followers as participants in hate groups defined by their inability to accept homosexuality and religious minorities.
Left-wing followers are more likely to protect the rights of the individual. for example, left-wing followers would tend to support same-sex partners and abortion.
I by no means am an expert in politics. This is simply my own understanding of what how those who follow each wing are loosely defined.
This will be dependent upon context as other editors have pointed out, and so my answer as a Brit might be different from the answers that other editors might give you. However, broadly speaking right-wing parties and policies try to promote private ownership and enterprise and conservative approaches to government and the economy, whereas left-wing parties stand for more liberal approaches and in particular being kinder towards the have-nots of society. Right wing parties are perhaps lambasted unfairly as being for the preservation of the haves of society and their privileged position. Left-wing parties show more of a social conscience and try to challenge inequalities.
One of the differentiating factors between the political left and right in American government is that the (1) right believes in a smaller government and--in recent years--with more power but less economic oversight and regulation. The (2) left believes in an encompassing government that offers aid to citizens and participates in active economic oversight and regulation. Think (1) George W. Bush and the Clinton impeachment attempt for the right and think (2) Franklin Delano Roosevelt and his economic policy for the left.
The way I was always taught the difference as a child is to remember the saying,"Give a man a fish and he'll eat for a day. Teach a man to fish and he'll eat for a lifetime."
The point is that those on the left are perceived as people who support entitlements (or gifts) to those who are less fortunate so that the "wealth can be spread" more evenly among the people.
Whereas the right tends to support individualism, or allowing the people the freedom to earn for themselves what they need to survive and thrive.
The answers above have all given you great descriptions of the right and left wings of the government. In the United states these are pretty much separated by Democrats and Republicans. In my opinion, these two "wings" of government are going to have to come together and make decisions based on what is best for the people and not what is best for their party.
I think this is an interesting time for Republicans and Democrats. At this time, each is struggling to redefine itself around job creation. Republics think that if you give more money to the rich, they will create jobs. Democrats, on the other hand, want to create jobs for the poor.