Interpolation is the idea that, with a set of known values, you can estimate the values of additional points based on their proximity to these known values. This recipe shows you how to use known values at point locations to create a continuous surface (raster) of value estimates. Classic examples include weather data estimations that are based on weather station data (think temperature or rainfall maps), crop yield estimates that are based on sampling parts of a field, and like in this example in this recipe, elevation estimations that are based on the elevation of sampled points.
Activate Interpolation Plugin via Plugin Manager.
Load a point layer with numeric columns, representing the feature of interest. For this recipe, use the poi_names_wake.shp, and the elev_m column, which contains elevation in meters for each point.
poi_name_wake.poi_names_wake for Input.elev_m for Interpolation attribute.100 and 100. This forces the output cells to be 100x100 units of the current projection.Generally, if this was for analysis, you would attempt to match the region of interest or other raster layers. In this case, we just want to go for sensibly-sized cells. As the map is in UTM, we will want cells to be integers that represent metric units; 100 meters by 100 meters makes interpreting the results easier.
idw100m (the result will be an ASCII raster .asc file), as shown in the following screenshot:

The basic idea is that, at a given cell, you take the average of all the nearby points that are weighted by their distance to the cell in order to estimate the value at your current location. Inverse Distance Weighted (IDW) takes this one step further by giving more weight to values that are closer to the given cell and less weight to values that are further. This function uses an exponent factor P in order to greatly increase the role of closer points over distant points.
Are the results not quite what you expected? There are a few parameters that can be adjusted; these are primarily the P value and the size of the cell. Is this still not coming out the way that you want? There are several other Interpolation tools that are accessible in Processing under the SAGA, GRASS, and GDAL toolboxes, which allow you to manipulate more of the formula parameters to refine the results.
Finally, depending on your data, IDW may not do a good job of interpolating. In the example here, you can actually see how there are distinct circles around isolated points. This is generally not a good result, and this needs a smoother transition to nearby points. If you have any control over field sampling to begin with, keep in mind that regularly-spaced grids will usually provide better results.
Do you not have control over the source data or you didn't get good results? Then, you may need to look into other more complicated formulas that compensate for skew, strong directionality, obstructions, and non-regular spacing of samples, such as Splines or Kriging, or Triangulated Irregular Networks (TINs). There is lot of science and statistics behind the methods and diagnostic tools to determine the best parameters. This is far too complicated a topic for this recipe, but it is well-covered in books on geostatistics.