Due to the fact that a projection changes three-dimensional reality into a two-dimensional drawing of that reality, something has to give. There will always be distortion when moving from a three-dimensional world to a two-dimensional world. There are six types of distortion which any given map might show (based on its projection) and every two-dimensional map will have three or more:
Area distortion means that the total land area of a country on a world map may be more or less than it actually measures on the ground.
Shape distortion means that the shape of a country on a world map will be distorted from its actual shape.
Direction distortion means that the map cannot be used to navigate in a straight line from one point on the map to another.
Bearing (or direction) distortion means that a straight line drawn between two points on the map will not have the same bearing as when using a compass to navigate between those two points in the real world.
Distance distortion means that calculating the distance on the ground from two points on the map (using the map's scale) would not be accurate.
Scale distortion means that the map scale is only valid at the points where the flat map projection intersects with the sphere of the Earth. Within that circle of intersection, the scale will be too small. Outside of that circle, the scale will be too large.
Depending on the purpose of the map, some distortions are acceptable and others are not. One map projection may preserve bearing and shape while distorting area and distance. Another might be just the reverse. There are many different combinations in varying degrees which result in the hundreds of projections in use today.
This multitude of map projections exist in order to preserve some of the above properties of the sphere while sacrificing three or more of the other properties. The characteristics of any particular projection make it appropriate for certain uses and inappropriate for other uses.