Windows can read and write to NTFS and FAT, with some support for other filesystems if you install the appropriate software. Ubuntu, on the other, can read and write to ext2, ext3, ext4, NTFS, and FAT.
So that means that out-of-the-box Windows and Ubuntu only have NTFS and FAT as filesystems in common. FAT is not good enough for Ubuntu, as it doesn't keep file permissions, and it's quite old and limited any way. NTFS is also not good enough for Ubuntu. Although it supports case-sensitive filenames, it does not support UNIX-like file permissions. This rules out installing Ubuntu on an NTFS filesystem (see this question).
That just leaves these options:
Use the NTFS or FAT partition devoted to Windows to keep the files you want to share between Ubuntu and Windows.
Create a third partition formatted as NTFS with no operating system, solely devoted to storing the files you want to share between Ubuntu and Windows.
Install some software to allow Windows to read the filesystem used by your Ubuntu installation (such as ext4). There are several ways to do this, and none of them are straight-forward.