When my king died, the next king (oldest son) could only raise like half the army (levies) I could before. What happened?
I think the reason your levy decreased is due to your heir's vassal-leige relationship score being different than your original king's. It could also be that your levy base providing buildings are not in your heir's personal demense.
The levy system is actually kinda complex but once you have a feel for it I suspect it'll make sense.
I originally said (in comments) that I thought this was related to titles, and it's not quite right. There's a few things at play.
If you don't give your sons any land then you can end up with the "unlanded sons prestige penalty". You need to give your first son some lands (but not the 2nd or 3rd as it'll increase a chance of war on inheritance). This shouldn't affect your levy though, except as it may indirectly affect your relationship with your vassals.
I came across a pretty decent description of the levy system [here].1
If you lose any of the following, you'll see a bad drop:
Here's where you could be losing levy... especially note that if the castle isn't in your personal demesne you'll take a hit.
The following is what I was talking about with the V-L Relation
Hope this helps and good luck! Sorry about any confusion!
Since you don't mention whether it's your personal levies or your vassal's levies...
Your ruler's martial skill also makes a difference in the the number of levies you can raise. If your deceased father had a martial of 20+ and your new whelpling has a martial of 10 or less, you'll see a significant difference. Even though you always raise 100% of the available levy in any of your demense holdings, what that figure is changes based on your personal martial skill (and buildings built and county modifiers). Quoting from the CKII wiki:
So if you went from 20+ to 0 martial skill, you would go from 150% to 50% personal levies, which is a 2/3rds reduction. If you have a large demesne and a low vassal count, this could be responsible for the bulk of the difference.
Also, the military tab (sword & shield icon, right of the ruler portrait) can show you what proportion of your levies come from your Demesne vs. your Vassals.
Below that part, there's a section that breaks down your vassal levies. If you hover over the green/red bar split, you get a summary of the math for that vassal. This should look roughly similar to the math in Tharius' excellent answer.