There are at least a few different ways in which to answer this question.
To start with the obvious answer, births and immigration increase a country’s population. When a person is born or when a person moves into the country, the country’s population goes up. By contrast, deaths and emigration reduce a country’s population. When a person dies or leaves the country, there are fewer people in the country than there previously were and the population drops.
But there must surely be more to this question. Another thing that we can say about births is that they add to the bottom of the country’s population pyramid. A population pyramid shows what percentage of the country’s population is of various ages. Each birth adds to the lowest level of the pyramid. Deaths are not quite so consistent. In general, older people are more likely to die, so most deaths will tend to thin out the upper levels of the country’s population pyramid.
Immigration affects population in most countries in more than one way as well. Most immigrants coming to countries in the rich world tend to be relatively young. Thus, their entry increases the size of the lower levels (though not usually the lowest levels) of the population pyramid. In addition, immigrants to many rich countries tend to come from cultures that typically have higher fertility rates. This means that immigration often means that the country’s population will go up in the short term (because the immigrant has entered the country) but also in the longer term, because the immigrant will have more children on average than a person who is native to the country. Since emigrants tend to be relatively young, emigration tends to hollow out the lower to middle regions of a country’s population pyramid. If many people in the 20 to 35 year age range leave a country for better economic opportunities elsewhere, the population pyramid narrows a little in the middle, leaving somewhat more of an hourglass shape as more of the population is either relatively young or relatively old.
In these ways, births, deaths, immigration and emigration can all affect both the size of a country’s population and its distribution along the vertical axis of a population pyramid.