The two major sticking points at the moment (October 2017) are:
The UK has made a basic offer to give EU citizens already in the UK a fast track permanent residency application service. So far it has been a bit of a fiasco, initially requiring an 85 page form to be filled out and being rather expensive. On top of that the Home Office has been pushing hard to make people leave the country, resulting in letters threatening deportation being sent in "error".
The EU wants citizens to retain their existing rights. That means, for example, that they can leave the UK and come back at a later date. The current UK offer terminates if you leave for too long without obtaining full citizenship. Similarly, UK ex-pats are concerned that they will be stuck in the EU country they are in now, losing the right to move to a different one for work.
Family members are also a major unresolved issue. Freedom of movement allows family members, even non-EU ones, to join the person exercising their treaty rights. This bypasses the local immigration system entirely. Failure to maintain existing rights would rip families apart.
There is also the issue of jurisdiction. The UK is adamant that the European Court of Justice (ECJ) can no longer have control after Brexit, but the EU is unwilling to allow this because the UK courts are already failing to comply with agreed EU rules. For example, the current treatment of non-EU family members is under active investigation.
The UK has many made financial commitments. As well as agreeing the EU budget up to at least 2021, it is committed to funding various EU projects. Pensions for EU staff (including British staff), funding for development projects, funding for licencing and regulatory authorities, research and more have all been agreed to.
The EU requires a commitment in principal to honour these commitments, with the exact amount to be calculated later but thought to be around €60bn. The UK has committed only to the EU budget during a transition period, not all the other stuff.