How does population affect my planets? Influence?

Dedicate tiles to as much pop as you can keep happy.  Population is the prime determinate for Raw Production.  It weighs across the board whereas bonuses to man/ res/ wea only effect the portion of Raw Production assigned to them.

Influence works better scattered, though focusing on a planet near where you're trying to culture flip a rival certainly helps.

Population provides the raw numbers for production, research, money, etc. The facility (labs, factories, econ buildings) add percentages including both a basic level and an addition for each level of building. Power plants, the database and other special additions like the associated capitol (manufacturing, research and financial) add to both the base number and percentage. You are taking the output of the populace and using the buildings as force multipliers.

Building doesn't cause unhappiness; increasing your population does: buildings do increase maintenance costs. A single point of production from population be worth 5 or more of research or production due to the building multipliers.

I prefer to have one or two influence buildings on most planets, concentrating more on planets with a bonus in that area. If you have the influence tool tip on, you can see that the spread is from a planet producing influence, dropping off as you get farther away. This is true if you move the cursor between two of your own colonies.

Thank you for the replies. How many pop tiles should be allocated for a research focused planet?

About as many as lab tiles if you have the ability to keep the populace happy, as a general rule of thumb. Exact numbers depend on too many variables for there to be a single answer that works optimally for all possible research-focused planets; things like empire-wide research bonuses can skew things in favor of more population tiles, while approval issues and time required for the population to grow can skew things back towards research tiles. Even planet tile layout can skew things through the effects of adjacency bonuses, if you're looking for optimal planet builds.

I don't look at it as a balance labs vs pop.  I look at it as keeping my population growing towards a pop cap and keeping that pop cap ahead of the growth.  I generally start with a hospital and two farms on every planet, whatever infrastructure keeps the populace happy and productive, and the the rest is the planet specialization, in this discussion, labs.  Late game, I add a one or two pop tiles through terraforming depending on what growth rates I have achieved.  And again, everything else pumps up the specialization.  This approach seems to work for both high and low tile counts.  It has to be adjusted for precursor worlds and such and that is a case by case issue for me.

To summarize: get as much production as you can (people), and then put that production through as many specialization bonuses as you can (labs).  That's my excuse for a plan and I'm sticking to it!