"Soviet" means something like "council" or "committee" in Russian. It's related to the word for advice. Popular councils played an important role in the Russian Revolution and were supposed to provide the foundation for a new type of worker's state, so the Soviet Union is the Union of Councils. The various "Soviet republics" that sprung up after 1918 were supposed to be unions of councils in the same way -- the Räterepublik Baiern is the Bavarian Republic of Councils, the Slovenská republika rád is the Slovak Republic of Councils, etc.

"People's republic" refers to the idea of people's democracy -- which was, depending on how you think about it, either a cynical scheme to expand Communist influence and Soviet power without people noticing, or an alternative route to communism specific to the period after the Second World War. The idea was that in Eastern Europe and Asia, Communists had worked within broad national coalitions to defeat fascism and imperialism, and that this struggle laid the foundation for socialist development without the need for violent revolution or the establishment of a dictatorship of the proletariat. Big industrialists and fascist collaborators (proponents of people's democracy basically understood these two groups as basically the same thing) would be expropriated and excluded, but other non-proletarians -- like peasants, small business owners, and the intelligentsia -- would be allowed to participate in building socialism, and parties representing them could continue to exist. Eventually, however, most Communist leaders largely away from the idea of people's democracy as a different path to socialism and began pursuing policies that established one-party states along the Soviet model.

Yugoslavia, Poland and Hungary officially declared themselves People's Republics when they adopted new constitutions in 1946, 1952 and 1949. Although Czechoslovak leaders absolutely considered their country to be a people's democracy and talked about people's democracy all the time, they never added it to the official name, even though it also adopted a new constitution shortly after the Communist seizure of power in 1948. Instead, they waited until 1960 to pass another constitution, and declare that they had achieved socialism and could now call themselves the Czechoslovak Socialist Republic. Yugoslavia also renamed itself the Yugoslav Federated Socialist Republic in 1963.

Regarding the term "soviet" - it's a bit of an anachronism and mistranslation. The Russian word specifically refers to workers' councils, and so at least in the case of Hungary and Bavaria, similar revolutionary socialists who briefly took power also instituted "council" republics in their respective language. But as far as I am aware, the Hungarian and Bavarian socialists of the time at least didn't explicitly use the term "soviet", as that was a Russian term, and using this in the English translation is more to draw direct connections with the Russian Bolsheviks.

As for "People's Republics" - that's not really just a rebrand of the same thing that people were trying to do in 1917-1921. The idea of a People's Republic came out of the Yalta and Potsdam Agreements, where the USSR committed to allowing free and fair elections to be held in Eastern European countries, both occupied Axis ones (like East Germany, Hungary, Romania and Bulgaria) and Allied ones (like Poland and Czechoslovakia). The Soviets expected local Communist parties to win elections and take power through parliamentary means. This generally didn't happen as they expected (also Czechoslovakia came closest), and so the People's Republics were something of a fudge - Communist Parties pushed through mergers with the (usually much larger and much more popular) Socialist Parties, and then these super-parties in turn led Popular Fronts formally comprised of other ideological parties that in reality did anything the Communist-controlled party told them to (but nevertheless served roles to mobilize and inform certain sectors of postwar society).

So while the Bavarian Council Republic was much closer to the 1917 Bolshevik model comprised of revolutionaries mostly from the Independent Socialist USPD who mostly just held power in Munich for less than a month, the German Democratic Republic (DDR) was ruled (with the implicit and explicit support of occupying Soviet military forces) by a Socialist Unity Party formed out of the region's former Communist and Socialist Party, and leading a National Front that also contained junior party members, such as a Christian Democratic Party, a Liberal Democratic party, an agrarian one and even a nominally far right (!) party. Other Eastern Bloc countries had a similar model, as do China and North Korea to this day. The USSR was actually almost unique as the "first" socialist state in that it constitutionally only had a Communist Party as its sole legal party and guiding force for the state.