ssch's answer is very helpful. However, there is another possibility to introduce a length scale (which I prefer), namely
VertexSize -> {"Scaled",.1}
as described by the documentation article VertexSize
.
This determines how large a vertex is given in units, where the "overall diagonal" of the graph equals 1.
Many of the times, this is very similar to Scaled[...]. However, Scaled[...] has the "possible issue" (see documentation of Scaled
) that if your graphics has an AspectRatio that does not equal 1, the nodes of a graph get stretched.
For instance, compare the two outputs of
Graph[{1, 2, 3}, {1 <-> 2, 2 <-> 3}, VertexSize -> Scaled[0.2]]
Graph[{1, 2, 3}, {1 <-> 2, 2 <-> 3}, VertexSize -> {"Scaled", 0.2}]

Sadly, if you desire graphs of different "overall diagonal" to have the same vertex size, you will still have to rely on the solution of Sjoerd C. de Vries.
For simple graphs, a very good option is to use your taylor made VertexShapeFunction. However, be sure that you also correctly constrain the PlotRegion! The following code exemplifies this approach. Note the trick to get rid of the nasty superfluous padding (white space) around a graph (as apparent in of Sjoerd C. de Vries' solution).
graphRange[graph_, \[Delta]_] :=
Block[{vCoords, minX, maxX, minY, maxY},
vCoords = (VertexCoordinates /.
AbsoluteOptions[graph, VertexCoordinates])\[Transpose];
minX = vCoords[[1]] // Min;
maxX = vCoords[[1]] // Max;
minY = vCoords[[2]] // Min;
maxY = vCoords[[2]] // Max;
{{minX - \[Delta], maxX + \[Delta]}, {minY - \[Delta],
maxY + \[Delta]}}
];
graphSize[graph_, \[Delta]_] :=
Block[{range, sizeX}, range = graphRange[graph, \[Delta]];
sizeX = range[[1, 2]] - range[[1, 1]]
];
ShowGraph[g_, scale_, vertexSize_] :=
Show[g, PlotRange -> graphRange[g, 2 vertexSize],
ImageSize -> scale * graphSize[g, 2 vertexSize]];
vertexSize = 0.2;
vShape[{xc_, yc_}, name_, {w_, h_}] := Disk[{xc, yc}, vertexSize];
g1 = Graph[{1, 2, 3}, {1 <-> 2, 2 <-> 3},
VertexShapeFunction -> vShape];
g2 = Graph[{1, 2, 3}, {1 <-> 2, 2 <-> 3, 1 <-> 3},
VertexShapeFunction -> vShape];
scale = 100;
ShowGraph[g1, scale, vertexSize]
ShowGraph[g2, scale, vertexSize]


FullForm
. $\endgroup$