Gaining a Ducal or Kingly title can be a great way to further your realm’s expansion. Once you have acquired the title, you also acquire a claim on any lands that are part of that de jure Duchy or (in patch 1.05) Kingdom, but not yet a part of your realm. For an example, let’s use the de jure map of Ireland again.
From the size of the Coat of Arms, we know that there are Dukes in Connacht and Munster. We can also tell from the borders that these Dukes do not own all of the de jure lands in these Duchies. In Connacht, Breifne is independent, and in Munster, Desmond is independent. In both of these situations, we know that the owners of the Duchies have de jure claims on the owners of the independent counties, and can declare war at any time with success forcing the Count into vassalage. This de jure claim exists for any Duchy or Kingdom which is not completely owned by the same independent realm. When your realm can press a de jure claim on another realm, you will see a message alert button in the top center of your screen:
De jure claims are very powerful because they are immediate (once you have the associated title, you also have the claim), and eternal (as long as you have the title, you also have the claim). If you take two counties in Ulster and create the Ducal title there, you know that you can press a war on the remaining two counties whenever it is convenient.
The drawback to de jure claims is that they do not always add lands to your demesne. In the example of de jure claims in Ulster, pressing the claims merely subjugate the existing Counts, who will become your Vassals. As per Power in the comments, when winning de jure wars against infidels, you (if you’re a duke), or your vassal (if you’re king), will directly annex the province bringing it into your/their demesne, but mayors will remain of the old faith. [Thanks, Power! -Ryan]
A properly planned out expansion will take into account possible de jure claims, and how best to use them to get both the Realm and Demesne that will insure a powerful and lasting Dynasty. However, it is also important to pay attention to which Duchies and Kingdoms have de jure claims on your lands. If you start as the Count of Breifne above, the de jure ducal claim that Connacht has on you from day one should factor heavily in your decision making.
Other De Jure Topics:
3/23 Fixed wrong icon for de jure ducal claims alert, doh!