You summarize when you relate to your reader what another person said or wrote; you analyze when you evaluate the validity or quality of another person's ideas. Suppose you had a long conversation with a friend about recent cuts to student pay rates. You might later tell your roommate that your friend believes the pay cuts are harmful to students. That's a summary of your friend's ideas. If you go on to say that your friend's ideas on the issue are not trustworthy because she holds two campus jobs, you are analyzing. You expressed your opinion or assessment of someone else's ideas instead of just repeating those ideas.
In academic papers, some summary is necessary to help orient readers and prepare them to understand your argument. However, simple summary is rarely enough to satisfy the conditions of your assignment. You will likely have to analyze the ideas you summarize.
Here's an example of a paragraph summarizing historical facts and an author's ideas:
Las Casas attacks Spain's nominal attempts to legitimize murder and pillage, like the Requerimiento (1513). This regulation required that Spanish colonials read a decree to native peoples under attack, offering them the opportunity to accept Christianity and become vassals of the King of Spain. Otherwise, war was justified, and natives could be deprived of their property and autonomy. However, las Casas argues, the text of the decree was often recited far outside of the area under siege, or without a native translator present. Las Casas decried the abuse of the law and even belittled the concept behind it, calling the idea of instant or forced conversion "as absurd as it is stupid" (Las Casas, Short Account 33).
The writer isn't asserting that las Casas was right or wrong, or providing any interpretation of what he wrote. This is a presentation of evidence, not analysis of it.
Here's an example of a paragraph analyzing and evaluating las Casas' stance:
Although las Casas objected to key aspects of Spanish colonialism, he did not go so far as to challenge the enterprise itself. His protests were limited to the manner in which the Spanish colonized. Indeed, las Casas was a strong advocate of the Christianization of the native peoples of Latin America; one of his major grievances with Spanish colonization was that it undermined attempts at conversion. Las Casas placed the obligation to facilitate evangelization on the Spanish crown and society; in his eyes, the Spanish were best suited to the responsibility of spreading the faith to the people of the Americas. Likewise, his arguments against war, theft, enslavement, and murder simply indicated that he rejected the practices of Spanish settlers. He did not deny their right to inhabit and rule the New World, provided it was properly done. Even if he adopted this stance to please those in power, he did not broaden his argument to question Spain's authority in the Americas. His radical defense of the native peoples attacked Spanish practices, not the entire colonial order.
Interpretation is a third way to relate to another person's ideas. When you are interpreting primary sources—like a painting or a set of historical documents—then your interpretation will be more like analysis than summary. This is because the question raised by primary sources is precisely "what should we make of this?" and so you will have to defend, with argument, whatever answer you give.
By contrast, when you're interpreting secondary sources—arguments that other people have made about primary sources—then your interpretation should be more like summary. Since the ultimate goal is to get at a better understanding of the primary sources, the purpose of interpreting secondary sources is to show options for how one might understand a primary source. Once you have summarized those options, you can then start critically evaluating them in the analysis stage of the paper.