A Type 1 hypervisor runs directly on the hardware with no operating system underneath it. The hypervisor itself is the lowest layer of software. VMware ESXi, the open-source KVM and Xen, and Microsoft Hyper-V are the common ones. This is what every cloud provider runs, because there is no host OS stealing resources and the attack surface is smaller.
A Type 2 hypervisor runs as a normal program on top of an existing OS like Windows or macOS. VirtualBox, VMware Workstation, and Parallels are examples. It is convenient for running a Linux VM on your laptop, but it is slower because guest instructions pass through both the hypervisor and the host OS.
Under the hood, modern hypervisors lean on hardware support: Intel VT-x and AMD-V let the CPU run guest code at near-native speed and trap only the privileged instructions the hypervisor must handle. Intel EPT and AMD RVI handle memory address translation in hardware so the hypervisor does not have to shadow every page table. For devices, the hypervisor either emulates a virtual NIC and disk, or uses paravirtualized drivers (virtio on KVM) where the guest knows it is virtualized and talks to the host efficiently, or passes a real device straight through with SR-IOV and IOMMU.