A system of government in which elections are held and one of more genuine opposition parties are permitted to exist, but as a practical matter it is almost impossible for the opposition party to elect enough representatives to have meaningful influence in governmental policy making, is called a "dominant party system". These systems are also sometimes called "semi-democracies."
Some of the leading examples include Mexico from 1929-2000, during which the same party (the PRI) always won national elections, the Democratic Party in many U.S. states in the former Confederate States from the end of Reconstruction (in the late 1800s) until the early 1960s, Iran for a significant part of its history, and Russia for a significant part of its post-communist era. Another example is the Democratic Party in the State of Hawaii, where Democrats control all statewide elected offices, all federal elected offices, 24 of 25 seats in the state senate (after winning every seat in the state senate for a two year period), and 47 out of 51 seats in the state house, and have done so for many years. Many other examples around the world are identified in the link above.
Sometimes these countries transition from having a dominant party system to a competitive more than one party system gradually, with an emergent party gradually winning a few more minor seats in local governments and a few legislative seats until eventually it wins control of a local government or enough legislative seats to have the opportunity to form a legislative coalition with other minor parties. For example, this is what happened in the American South and in Mexico and what is happening to some extent in Iran.
Sometimes these countries transition away from having a dominant party system in a punctuated fashion, often due to a schism in the formerly dominant party that brings elected officials with it (sometimes after the collapse of another political party as in the case of the birth of the Republican party in the U.S.), or in an electoral movement suddenly mobilized for a single election based upon a celebrity candidate or with a social movement reacting to some galvanizing event in what had previously been a moribund opposition party organization.
Similarly, a galvanizing event or celebrity can sometimes cause a competitive political system to become a dominant party system. For example, the Great Depression led to this result in the United States with the Democratic party becoming the dominant party. In many countries (e.g. India and South Africa) the party the led the movement for the nation's independence initially becomes the dominant party.
Another situation that sometimes emerges is that a political system is nominally a one party system or a non-partisan system, but stable factions of members of the only legally permitted political party, or of non-partisan elected officials, that are comparable to political parties, emerge within this system and contest with each other for political power over the government or entity in question.
This is how political parties emerged in both France and the U.S., historically. This is also present, for example, in the internal politics of the Roman Catholic Church, and in the internal politics of the royal family in Saudi Arabia.