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The policy of containment was first announced on 12th March 1947 in Truman's speech to Congress which outlined its broad goals that would later be formalized into US foreign policy. The Truman Doctrine is the other name for the policy of containment since it was initiated by Truman and his administration.
A direct quote from the speech that outlines the intent, method and goals of containment can be seen from this extract:
To ensure the peaceful development of nations, free from coercion, the United States has taken a leading part in establishing the United Nations, The United Nations is designed to make possible lasting freedom and independence for all its members. We shall not realize our objectives, however, unless we are willing to help free peoples to maintain their free institutions and their national integrity against aggressive movements that seek to impose upon them totalitarian regimes. This is no more than a frank recognition that totalitarian regimes imposed on free peoples, by direct or indirect aggression, undermine the foundations of international peace and hence the security of the United States.
The peoples of a number of countries of the world have recently had totalitarian regimes forced upon them against their will. The Government of the United States has made frequent protests against coercion and intimidation, in violation of the Yalta agreement, in Poland, Rumania, and Bulgaria. I must also state that in a number of other countries there have been similar developments.
At the present moment in world history nearly every nation must choose between alternative ways of life. The choice is too often not a free one.
One way of life is based upon the will of the majority, and is distinguished by free institutions, representative government, free elections, guarantees of individual liberty, freedom of speech and religion, and freedom from political oppression.
The second way of life is based upon the will of a minority forcibly imposed upon the majority. It relies upon terror and oppression, a controlled press and radio; fixed elections, and the suppression of personal freedoms.
I believe that it must be the policy of the United States to support free peoples who are resisting attempted subjugation by armed minorities or by outside pressures.
I believe that we must assist free peoples to work out their own destinies in their own way.
I believe that our help should be primarily through economic and financial aid which is essential to economic stability and orderly political processes.
The world is not static, and the status quo is not sacred. But we cannot allow changes in the status quo in violation of the Charter of the United Nations by such methods as coercion, or by such subterfuges as political infiltration. In helping free and independent nations to maintain their freedom, the United States will be giving effect to the principles of the Charter of the United Nations.
So lets first break down the context. The policy of containment was very much in response to the Soviet Unions use of salami tactics to influence elections in Eastern Europe and install pro-soviet governments. This spilt over into the Balkans and Turkey since they were undergoing their own civil war between the communist and democratic (At least nominally democratic, some of these elements were simply anti-communist) factions. This was set to the backdrop of increasing superpower tensions and mistrust and the Cold War was developing. The central objective of the USSR in the post-war world was to establish buffers between itself and potential aggressors and ensure that German remained an impotent threat to them. This involved establishing a sphere of influence primarily in Eastern Europe made of states friendly to Moscow which normally entailed that they were communist. In some states this was not as forced as in others as communism did have substantial legitimacy in Eastern Europe as the Red Army had liberated them for Nazi occupation and did see a good deal of popular support. That being said, in countries such as Czechoslovakia and Poland salami tactics were required in order to install a pro-soviet communist government due to the elections resulting in a coalition government made of other parties with other ideological leanings. All of this alarmed the US prompting a response as they saw this as Soviet expansionism.
The US was also competing for influence in Europe as it sought to establish a new post-war world order. Whether this is an idealistic dream of a future free from war or to establish US economic dominance is up to interpretation but suffice to say that they were looking for trade partners in Europe. Their alliance with the USSR was out of convenience with both Churchill and FDR along with a majority of the Allied leadership being sceptical of or outright anti communist. Now, this was beginning to show with the US and the UK distrusted the intentions of the USSR and believed that the spread of communism was detrimental to the Western way of life. Thus in order to curb this perceived Soviet expansionism, they initiated the policy of containment where they would support any factions fighting against the communist in an effort to curb the expansion of the Soviet sphere of influence.
Truman and the US leadership at the time did fear the domino effect would occur. This is known as Domino theory which posits that if one country fell to communism its neighbours would fall in short order causing a spread of communism throughout a region. Whether this holds true in reality is debatable but the important thing is that the US did believe it was true and thus were compelled to prevent it. Thus the policy of containment would serve to prevent this.
Whether this is a success or failure is hard to define since the Truman doctrine underpinned US Cold War foreign policy. The US had managed to successfully contain the spread of communism in instances such as Korea. However, they also faced failures in instances such as Cuba and Vietnam where communism successfully established itself there. There is also an argument that could be made for containment still being successful as the rest of region did not fall to communism indicating that it was contained to a few countries. The Truman Doctrine was later supplanted by the Reagan Doctrine which was the policy of rolling back communism. The Reagan Doctrine mandated the US pushback communism and democratize them instead of simply being content to contain them. This complicates the issue as the Truman doctrine towards the end of the Cold War in the 80s was effectively superseded which makes assessing it's success or failure difficult.
Overall you have the right basic idea of the Truman doctrine and its intended purpose.
Sources:
PRESIDENT HARRY S. TRUMAN'S ADDRESS BEFORE A JOINT SESSION OF CONGRESS, MARCH 12, 1947, The Avalon Project
The Tragedy of American Diplomacy by William Appleman William