Inequality within a country, poor or rich, can be remedied effectively through an income tax system- by taxing the rich and redistributing the resources directly or indirectly to the poor. Perhaps surprisingly, many rich countries have less inequality than poor countries partially because the poor in these countries are relatively rich compared to the poor in other countries and partially because votes in rich countries generally elect to take a larger fraction of income from the rich and give to the poor directly or indirectly. 
A popular measure of inequality in all countries is called the Gini index, named after its author Corrado Gini. This index effectively ranks individuals from poorest to richest and compares each individual’s rank to his or her share of total income.  If all income is distributed equally in a population, the Gini index is 0. If all income is earned by only one person and all others earn nothing, then the Gini index is 1. Many European countries have some of the lowest indices, between 0.25 and 0.30, while many African and South American countries have some of the highest values, between 0.55 and 0.65.  South Korea has a relatively low value of 0.31. 
International inequality usually refers to the fact that some countries are poor, while others are rich. The real issues, then, are why does poverty exist in some countries and how can these countries  grow out of poverty.  
The measure of poverty that economists use is the level of production of marketable good and services that can be sold, or the level of income earned from those sales, or the level of the expenditure on the goods and services bought.  
No one, including economists, has a foolproof explanation for why poverty exists or a magic formula to allow countries to produce and to sell more marketable goods and services, to earn more income, and to be able to afford to buy more marketable goods and services.  But experience has taught us that there are a few policies that seem to work more than others. 
Investments in education and health seem to be especially beneficial (needed) for poor people in poor countries, although there is a debate about what kinds of education and health are most beneficial or needed. Investments in roads, trains, ports, electricity, and clean water are useful to move goods and services to and from markets and to move people (that is, workers) to and from places of employment. But of course, these investments are very expensive, technically difficult and not always successful. When they could afford them, governments have often tried to make these investments. But these investments in many poor countries too frequently failed due to corruption and faults in engineering. Governments have sometimes invited private firms to make these investments instead, often with loan guarantees, subsidized credit or other government assistance.
Other government policies might seem less obvious, but can be necessary or very useful for an economy to function properly. Because market transactions often involve financial or other contracts, courts within the country need to be able to enforce these contracts.  This is harder than it might sound, especially in countries where corruption is a way for court officials to earn a higher income. 
As mentioned briefly before, governments can also provide loan guarantees or subsidized credit for private firms that want to invest in various industries.  India, China, and Brazil already do this with some success. But failures also result when the guarantees and subsidies given become an incentive for people to waste funds, making taxpayers pay for the loss. 
Finally, governments can remove barriers to international trade to expand markets and to allow domestic buyers to save money and often to get better quality products from foreign firms. Much evidence shows that eliminating trade taxes and requirements, which are intended to help import-competing firms, ends up hurting other institutions and individuals in the domestic economy even more.  And allowing free trade is especially beneficial when corruption exists among the government officials who regulate trade. 
Poor countries have complained that the US and the EU heavily subsidize some agricultural products and then dump these products in international markets.  While these complaints are justified, poor countries should instead focus on liberalizing their own markets, allowing people to produce, sell, travel and buy freely internationally.