What are costs associated with inventory? Why is controlling turnover in the inventory important? How can improvements in inventory management impact profitability?

Well, inventory is a combination of raw material,labor and over head; all the three transfer to Work In Process, and WIP transfer to Finish Goods Inventory.
The is anything from any of the process as describing above.
There is conflicting goal within an organization, the cost can be very complicated to determine, it's the work of purchasing department,finance department,product department.
Inventory turnover shows how fast/slow the inventory move on compared with previous years.
Improvements can be done based in all three departments I have listed, working close together try to avoid any "bottleneck", and increase best usage if time.

First learn about motives
MOTIVES OF HOLDING INVENTORY:
A Company may hold the inventory with the various motives as stated below:
a) Transaction Motive: An important reason for maintaining inventory levels is the transaction motive. The Company may be required to hold the inventory in order to facilitate the smooth and uninterrupted production and sale operations. It may not be possible for the company to procure the raw material whenever necessary. There may be a time lag between the demand for the material and its supply. Hence, it is needed to hold the raw material inventory. Similarly, it may not be possible to produce the goods immediately after they are demanded by the customers. Hence, it is needed to hold the finished goods inventory. The need to hold work-in-progress may arise due to production cycle.
b) Precaution Motive: Precautionary Motive for holding inventory is to provide a safeguard or cushion in case the actual level of activity is different than anticipated. In addition to the requirement to hold the inventory for routine transactions, the company may like to hold inventory to guard against the risk of unpredictable changes in demand and supply forces e.g. the supply of raw material may get delayed due to factors like strike, transport disruption, short supply, lengthy processes involved in import of raw material etc. Hence, the company should maintain sufficient level of inventory to take care of such situations. Similarly, the demand for finished goods may suddenly increase (especially in case of seasonal type of products) and if the company is unable to supply them, it will make a gain to competitors. Hence, company will like to maintain sufficient supply of finished goods.
c) Speculative Motive: It refers to the desire of a firm to take advantage of opportunities of rising prices which present themselves at unexpected moments and which are typically outside the normal course of business. While precautionary motive is defensive in nature i.e. firms keep inventory to meet unexpected contingent situations, the Speculative Motive represents an aggressive approach to exploit profitable opportunities. Advance purchase of raw materials in inflationary times is one form of speculative behavior.