In general, MPLS VPN is a service provider technology where the customer trusts the service provider to securely handle and deliver the traffic between sites. Layer 2 and Layer 3 MPLS utilize the same basic principles for transporting traffic across the provider network, namely MPLS label switching, setting up Tunnel Label Switched Paths (LSPs). However, the difference is that L3 VPN MPLS uses IP Routing and L2 VPN MPLS uses a circuit switching approach, similar to how leased line services were provided via frame relay or ATM using PVCs. The MPLS VPN approach is more flexible for service providers as it allows them to utilize the same networking equipment for all operations. The customer data is encapsulated and carried across the provider network via MPLS, label switched.
ADSL is just a telco DSL type. It describes how the connection to the customer premise is accomplished. There are many different ways to do this. It used to be frame relay, which is losing ground to now DSL, Gigabit Ethernet, fiber and others. There are mainly three components in every connection scenario: the provider network, the customer network and the last mile that bridges the connection between customer and provider. Sometimes this last mile piece can get tricky for providers to deal with due to the telco rights-of-way.
For more information on of MPLS VPN types, check out this MPLS VPN fundamentals guide.