A lectionary is pretty much a Bible reading plan that's followed by thousands of Christians (even from other denominations). It means we can all read the same Bible passages on the same days even though we can't all physically meet together.
There's a lectionary used by lots of denominations and across the whole world for the Bible passages that we'll all read 'together' on Sunday in church. It's called 'The Revised Common Lectionary'. (See my post on 'The Sunday Lectionary'.)
In the Church of England there's also the Common Worship 'daily prayer' lectionary. That one's for every day of the year - for Morning and Evening Prayer and midweek Eucharists.
I guess all reading the same Bible passages on the same days emphasises that we're one big Christian family and we're on a spiritual journey together. (NB what I said in a previous post about catholic spirituality being more 'we' than 'I'?) The lectionary helps us read through the whole Bible over a period of time and it also helps us focus on certain themes at certain times of the year. For example every day during Lent we read passages that focus on injustice and repentance and forgiveness. You can buy the 'daily prayer' lectionary as a little booklet from a Christian bookshop. There are also lectionary apps and websites.
I've found that one of the great things about following a lectionary in your personal prayer life is that you're not stuck deciding what Bible passage to read that day. Another great thing is it encourages you to read bits of the Bible you might not normally read, keeps your reading balanced by giving you an Old Testament reading, Psalm and New Testament reading each time, and keeps you from falling into the trap of just choosing the bits you know or like or that challenge you less! I like the fact that on any given day I can text a friend and say, "What did you make of this morning's Old Testament reading?!", or on Sundays I know that my friends in different churches around my city, in different towns across my country, and even my friends across the world in California are all reading and praying about the same Bible passages. It's quite common among my friends to have discussions on Facebook about 'this Sunday's Gospel reading'.
