Rogation days (coming from the Latin rogare, meaning "to ask") are days of fasting and prayer for the successful planting and harvesting of crops. They are held on April 25th, and on the Monday, Tuesday, and Wednesday leading up to Ascension Thursday. The major public celebration is a procession with set prayers and a recitation of the Litany of the Saints though at various points in history there were more elaborate practices. Traditionally during Rogation week, proceeding Ascension Thursday they would have a "beating the bounds" ( a procession lead by a priest around the boundary of the parish, prayers of blessing and protection are recited, and young boys would beat the ground with reads). They first appeared sometime around 470AD in Vienne and were officially adopted into the Roman Rite in the 7th century and introduced to the British Isles around the same time where they became even more elaborate. Ember days are three days (typically Wednesday, Friday and Saturday) set aside each quarter for fasting and prayer. They follow St. Lucy's feast day, the first Sunday in Lent, Pentecost, and the feast of the Holy Cross. Traditionally ordinations are commonly held on the Saturday Ember Day. They were traditionally days of abstinence and a procession in addition to the fasting and prayer.
I don't personally have enough in-depth knowledge of either to feel capable of describing them in my own words even though I do have a basic grasp of what both are. I have done some ferreting around on the Internet and cannot find anything completely to my satisfaction about them. However, I think the articles on both in Wikipedia are as good a starting point as any. Those articles may satisfy you or they may be a catalyst for you to do some more research. I don't think either of them are given much attention in any Western church these days. Both are also described in Encyclopaedia Britannica (EB). Neither article there is as in-depth as the Wikipedia ones plus the EB article of Rogation Days makes them out incorrectly to be only a Roman Catholic practice. Wikipedia on 'Rogation Days': https://en.wikipedia.org/wiki/Rogation_days Wikipedia on 'Ember Days': https://en.wikipedia.org/wiki/Ember_days EB on 'Rogation Days': https://www.britannica.com/topic/Rogation-Days EB on 'Ember Days': https://www.britannica.com/topic/Ember-Day-and-Ember-Week
They're both a fairly big deal at least here in the United States among the Anglican Continuum. They are also observed in the Ordinariates, among RCC Latin Mass communities especially the FSSP, as well as a growing number of diocesan parishes. There are also a number of religious orders (Marian Fathers, Trappist, Cistercians and some Benedictines to name a few) that observe them to some degree.